Linear Functions and Graphs Practice Questions — SAT

Question 1Question →

Two lines, $L_1$ and $L_2$ , are graphed in the $xy$ -plane. Line $L_1$ passes through the points $(2, 11)$ and $(6, 23)$ . If line $L_2$ is perpendicular to line $L_1$ and contains the point $(3, 10)$ , what is the $x$ -coordinate of the $x$ -intercept of line $L_2$ ?

Show answer & explanation

Answer: 33

Answer

33

To find the

x

-coordinate of the

x

-intercept of line

L_2

, first determine the slope of line

L_1

from the given points

(2, 11)

and

(6, 23)

using the formula

m = \frac{y_2 - y_1}{x_2 - x_1}

, which results in

m_1 = \frac{23 - 11}{6 - 2} = 3

. Since line

L_2

is perpendicular to line

L_1

, its slope must be the negative reciprocal of 3, which is

-\frac{1}{3}

. Using the point-slope form with the point

(3, 10)

, the equation of line

L_2

is

y - 10 = -\frac{1}{3}(x - 3)

, which simplifies to

y = -\frac{1}{3}x + 11

. Setting

y = 0

to find the

x

-intercept yields

0 = -\frac{1}{3}x + 11

, which simplifies to

x = 33

.

Step-by-Step Solution

1

Calculate the slope of line

L_1

using the points

(2, 11)

and

(6, 23)

with the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

.

m_1 = \frac{23 - 11}{6 - 2} = \frac{12}{4} = 3

The slope of a line represents its rate of change and is required to find the relationship with perpendicular lines.

2

Find the slope of line

L_2

which is perpendicular to

L_1

.

m_2 = -\frac{1}{3}

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Write the linear equation for line

L_2

using the point-slope formula

y - y_1 = m(x - x_1)

with the point

(3, 10)

and slope

m_2 = -\frac{1}{3}

.

y - 10 = -\frac{1}{3}(x - 3) \implies y = -\frac{1}{3}x + 11

Defining the equation of the line allows us to find the coordinates of any of its intercepts.

4

Set

y = 0

in the equation for line

L_2

to find the

x

-coordinate of the

x

-intercept.

0 = -\frac{1}{3}x + 11 \implies \frac{1}{3}x = 11 \implies x = 33

The

x

-intercept is the point where the line crosses the

x

-axis, which mathematically corresponds to

y = 0

.

Key Concept

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other. The

x

-intercept of a linear graph represents the value of

x

when

y = 0

.

Question 2Question →

A line representing the linear function $f$ is graphed in the $xy$ -plane. The line intersects the $y$ -axis at $(0, 3)$ and also passes through the point $(2, 11)$ . What is the slope of this line?

Show answer & explanation

Answer: 4

Answer

The slope of the line is 4.

The slope

m

of a line passing through two points

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

m = \frac{y_2 - y_1}{x_2 - x_1}

. Substituting the given points

(0, 3)

and

(2, 11)

into the formula gives

m = \frac{11 - 3}{2 - 0} = \frac{8}{2} = 4

.

Step-by-Step Solution

1

Identify two coordinates on the line from the given information.

The points are

(0, 3)

and

(2, 11)

.

To find the slope of a line, we need at least two points on that line.

2

Apply the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

.

m = \frac{11 - 3}{2 - 0}

The slope formula calculates the ratio of the vertical change (rise) to the horizontal change (run).

3

Perform the subtraction and division to find the slope value.

m = \frac{8}{2} = 4

Simplifying the fraction gives the final numerical slope.

Key Concept

Calculating the slope of a linear function given two points

Question 3Question →

In the $xy$ -plane, the graph of a linear function $f$ passes through the points $(0, 1)$ and $(2, k)$ . The graph of another linear function $g$ is perpendicular to the graph of $f$ and passes through the points $(-4, 8)$ and $(2, k)$ . Which of the following could be the value of $k$ ?

2

8

5

9

Show answer & explanation

Answer: 5

Answer

5

The slope of the line representing function

f

is calculated as

m_f = \frac{k - 1}{2}

, and the slope of the line representing function

g

is

m_g = \frac{k - 8}{6}

. Since the lines are perpendicular, the product of their slopes must be

-1

. This gives the equation

\frac{k - 1}{2} \cdot \frac{k - 8}{6} = -1

, which simplifies to

(k - 1)(k - 8) = -12

. Expanding and setting the quadratic equation to zero gives

k^2 - 9k + 20 = 0

. Factoring this expression yields

(k - 4)(k - 5) = 0

. Thus,

k

can be either

4

or

5

. Since 5 is the only option listed, the option containing 5 is the correct answer.

Step-by-Step Solution

1

Express the slope of the linear function

f

in terms of

k

.

m_f = \frac{k - 1}{2 - 0} = \frac{k - 1}{2}

The slope formula is

m = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Express the slope of the linear function

g

in terms of

k

.

m_g = \frac{k - 8}{2 - (-4)} = \frac{k - 8}{6}

The slope formula is applied to the points

(-4, 8)

and

(2, k)

.

3

Apply the perpendicular lines condition to set up an equation for

k

.

m_f \cdot m_g = -1 \implies \left(\frac{k - 1}{2}\right)\left(\frac{k - 8}{6}\right) = -1

Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is

-1

.

4

Solve the quadratic equation for

k

.

(k - 1)(k - 8) = -12 \implies k^2 - 9k + 8 = -12 \implies k^2 - 9k + 20 = 0 \implies (k - 4)(k - 5) = 0

Multiply both sides by 12, expand the product, move all terms to one side, and factor the quadratic expression.

5

Identify the possible values of

k

and match with the options.

k = 4

or

k = 5

. Since 5 is the only value present in the options, the correct value is 5.

Both values satisfy the condition, but only one is listed among the multiple-choice options.

Key Concept

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other (their product is

-1

).

Question 4Question →

A factory manufactures solar panels at a constant rate. At the start of a morning shift, the factory has already manufactured $80$ solar panels. The total number of solar panels manufactured $t$ hours after the shift begins is modeled by a linear function. If the factory has manufactured a total of $200$ solar panels $3$ hours after the shift begins, how many hours after the shift begins will the factory have manufactured a total of $440$ solar panels?

Show answer & explanation

Answer: 9

Answer

9

To find the time when the total number of solar panels reaches

440

, we first determine the constant rate of production. The change in solar panels over the first

3

hours is

200 - 80 = 120

panels. Dividing by the

3

hours gives a constant rate of

40

panels per hour. The linear equation representing the total panels is

N(t) = 40t + 80

. Setting

N(t) = 440

gives

440 = 40t + 80

, which simplifies to

360 = 40t

. Solving for

t

yields

t = 9

.

Step-by-Step Solution

1

Calculate the constant rate of production (slope) of the linear function.

The rate is

40

solar panels per hour.

The factory starts with

80

solar panels and reaches

200

solar panels in

3

hours. The rate of change is the change in the number of panels divided by the change in time:

\frac{200 - 80}{3 - 0} = \frac{120}{3} = 40

panels per hour.

2

Write the linear equation representing the total number of solar panels manufactured,

N(t)

, after

t

hours.

N(t) = 40t + 80

Since the initial quantity is

80

and the rate of production is

40

panels per hour, the linear function is

N(t) = 40t + 80

.

3

Solve for the time

t

when the total number of solar panels is

440

.

t = 9

hours

Set

N(t) = 440

in the equation:

440 = 40t + 80

. Subtract

80

from both sides to get

360 = 40t

. Divide both sides by

40

to find

t = 9

.

Key Concept

Linear Functions and Graphs

Estimated Time:1m 30s

Question 5Question →

In the $xy$ -plane, the graph of the linear function $f(x) = px + q$ , where $p$ and $q$ are constants, passes through the point $(2, 7)$ . Line $L$ is parallel to the graph of $f$ and has a $y$ -intercept that is $3$ units below the $y$ -intercept of the graph of $f$ . If the $x$ -intercept of line $L$ is $(-6, 0)$ , what is the value of $p$ ?

Show answer & explanation

Answer: 0.5

Answer

The value of

p

is

0.5

(or the fraction

1/2

).

The correct answer is

0.5

(or

1/2

). The graph of

f(x) = px + q

passes through

(2, 7)

, which means

7 = 2p + q

, or

q = 7 - 2p

. Line

L

is parallel to

f

, so its slope is

p

, and its

y

-intercept is

q - 3

. Thus, the equation of line

L

is

y = px + q - 3

. Since line

L

has an

x

-intercept at

(-6, 0)

, we can substitute

x = -6

and

y = 0

into its equation, yielding

0 = -6p + q - 3

. Substituting

q = 7 - 2p

into this equation gives

0 = -6p + (7 - 2p) - 3

, which simplifies to

-8p + 4 = 0

. Solving for

p

gives

8p = 4

, or

p = 0.5

.

Step-by-Step Solution

1

Express the relationship between

p

and

q

using the given point

(2, 7)

on the graph of

f

.

q = 7 - 2p

The point

(2, 7)

must satisfy the equation

f(x) = px + q

.

2

Formulate the equation of line

L

using the parallel slope and the shifted

y

-intercept.

y = px + q - 3

Parallel lines have equal slopes, and the

y

-intercept of

L

is

3

units below the

y

-intercept of

f

, which is

q

.

3

Substitute the

x

-intercept

(-6, 0)

into the equation of line

L

.

-6p + q - 3 = 0

The

x

-intercept is a point on the line where

y = 0

.

4

Substitute the expression for

q

from Step 1 into the equation from Step 3 and solve for

p

.

p = 0.5

(or

\frac{1}{2}

)

Solving the linear equation

-6p + (7 - 2p) - 3 = 0

simplifies to

-8p + 4 = 0

, giving

p = 0.5

.

Key Concept

Understanding linear functions, their graphs, slopes of parallel lines, and intercepts.

Alternative Method

Instead of solving for

q

first, you can use the point-slope form. Line

L

passes through

(-6, 0)

and has slope

p

, so its equation is

y = p(x + 6)

, or

y = px + 6p

. The

y

-intercept of

L

is

6p

. The

y

-intercept of the graph of

f

is

q

. We are given that the

y

-intercept of

L

is

3

units below the

y

-intercept of

f

, so

6p = q - 3

. Since

f

passes through

(2, 7)

, we have

7 = 2p + q

, which means

q = 7 - 2p

. Substituting this into

6p = q - 3

gives

6p = (7 - 2p) - 3

, or

6p = 4 - 2p

. Adding

2p

to both sides gives

8p = 4

, which results in

p = 0.5

.

Estimated Time:2m 30s

Question 6Question →

In the $xy$ -plane, the graph of the linear function $f$ has a slope of $-3$ and passes through the point $(4, 18)$ . What is the $y$ -coordinate of the $y$ -intercept of the graph of $f$ ?

Show answer & explanation

Answer: 30

Answer

The correct answer is 30.

The equation of a linear function can be written in slope-intercept form as

f(x) = mx + b

, where

m

is the slope and

b

is the

y

-coordinate of the

y

-intercept. Substituting the given slope

m = -3

yields

f(x) = -3x + b

. Since the point

(4, 18)

lies on the graph of

f

, we substitute

x = 4

and

f(x) = 18

into the equation to get

18 = -3(4) + b

, which simplifies to

18 = -12 + b

. Adding

12

to both sides of the equation yields

b = 30

. Thus, the

y

-coordinate of the

y

-intercept of the graph of

f

is

30

.

Step-by-Step Solution

1

Write the general slope-intercept form of a linear equation.

f(x) = -3x + b

The slope of the line is given as

-3

, so we substitute

m = -3

into

f(x) = mx + b

.

2

Substitute the point

(4, 18)

into the equation.

18 = -3(4) + b

Since the graph of

f

passes through

(4, 18)

, these coordinates must satisfy the function equation.

3

Solve for the

y

-intercept

b

.

b = 30

Simplify to

18 = -12 + b

, and then add

12

to both sides to isolate

b

.

Key Concept

Finding the equation of a line using its slope and a point.

Question 7Question →

The table below shows some values of the linear function $g$ .

$x$	$g(x)$
$2$	$5$
$4$	$11$
$6$	$17$

What is the slope of the graph of $g$ in the $xy$ -plane?

\frac{1}{3}

6

3

\frac{5}{2}

Show answer & explanation

Answer: $3$

Answer

The slope of the graph is

3

.

The slope of a linear function is the constant rate of change, calculated as the change in the function values divided by the change in the input values. Choosing the points

(2, 5)

and

(4, 11)

, the slope is

\frac{11 - 5}{4 - 2} = \frac{6}{2} = 3

. This same slope is obtained using any other pair of points from the table, such as

(4, 11)

and

(6, 17)

, where the slope is

\frac{17 - 11}{6 - 4} = \frac{6}{2} = 3

.

Step-by-Step Solution

1

Select two points from the table to find the change in the input and output values.

Using the points

(2, 5)

and

(4, 11)

, the change in the output

g(x)

is

11 - 5 = 6

, and the change in the input

x

is

4 - 2 = 2

.

The slope is defined as the change in the vertical coordinate divided by the change in the horizontal coordinate.

2

Divide the change in

g(x)

by the change in

x

to find the slope.

The slope is

\frac{6}{2} = 3

.

Dividing the vertical change by the horizontal change yields the constant rate of change of the linear function.

Key Concept

Finding the slope of a linear function from a table of values

Estimated Time:45s

Question 8Question →

The graph of a linear function is shown in the $xy$ -plane. The line passes through the points $(-2, -3)$ and $(2, 5)$ . If the point $(k, 11)$ lies on the line, what is the value of $k$ ?

9

14

5

6

Show answer & explanation

Answer: 5

Answer

5

The slope of the line is calculated as

m = \frac{5 - (-3)}{2 - (-2)} = 2

. Using the point-slope form with the point

(2, 5)

, the equation of the line is

y - 5 = 2(x - 2)

, which simplifies to

y = 2x + 1

. Substituting the point

(k, 11)

into this equation gives

11 = 2k + 1

, which solves to

k = 5

.

Step-by-Step Solution

1

Calculate the slope of the line using the points

(-2, -3)

and

(2, 5)

m = 2

The slope formula is

m = \frac{y_2 - y_1}{x_2 - x_1}

. Substituting the given coordinates:

m = \frac{5 - (-3)}{2 - (-2)} = \frac{8}{4} = 2

.

2

Determine the equation of the line using point-slope form

y = 2x + 1

Using

y - y_1 = m(x - x_1)

with the point

(2, 5)

and

m = 2

, we get

y - 5 = 2(x - 2)

, which simplifies to

y = 2x + 1

.

3

Substitute

(k, 11)

into the line equation and solve for

k

k = 5

Substituting

x = k

and

y = 11

yields

11 = 2k + 1

. Subtracting 1 from both sides gives

10 = 2k

, and dividing by 2 gives

k = 5

.

Key Concept

Finding the linear equation from coordinates and solving for variables

Question 9Question →

In the $xy$ -plane, line $k$ has the equation $y = -2x + 10$ . Line $m$ is parallel to line $k$ and passes through the point $(6, 8)$ . Line $l$ is perpendicular to line $k$ and intersects the $x$ -axis at the point $(a, 0)$ , where $a > 10$ . If the region bounded by lines $k$ , $m$ , $l$ , and the $y$ -axis has an area of $110$ , what is the value of $a$ ?

Show answer & explanation

Answer: 25

Answer

25

To find the value of

a

, we determine the equations of the lines

m

and

l

based on their geometric relationships to line

k

. Line

m

is parallel to line

k

(

y = -2x + 10

), so its slope is

-2

. Using the point

(6, 8)

, its equation is

y = -2x + 20

. Line

l

is perpendicular to line

k

, so its slope is

\frac{1}{2}

. It intersects the

x

-axis at

(a, 0)

, giving the equation

y = \frac{1}{2}x - \frac{1}{2}a

. The bounded region formed by the parallel lines

k

and

m

, the perpendicular line

l

, and the

y

-axis is a trapezoid. Calculating the area of this trapezoid by dividing it into a parallelogram and a triangle yields the area formula

60 + 2a

. Setting this equal to the given area of

110

yields

60 + 2a = 110

, which solves to

a = 25

.

Step-by-Step Solution

1

Determine the equation of line

m

using the parallel slope and the given point.

Line

m

has the equation

y = -2x + 20

.

Parallel lines have equal slopes. Since line

k

has a slope of

-2

, line

m

also has a slope of

-2

. Substituting the point

(6, 8)

into the point-slope form gives

y - 8 = -2(x - 6)

.

2

Determine the equation of line

l

using the perpendicular slope and its

x

-intercept.

Line

l

has the equation

y = \frac{1}{2}x - \frac{1}{2}a

.

Perpendicular lines have negative reciprocal slopes. The negative reciprocal of

-2

is

\frac{1}{2}

. Using the point

(a, 0)

in the point-slope form gives

y - 0 = \frac{1}{2}(x - a)

.

3

Calculate the vertices of the bounded region by finding the intersection points of the boundary lines.

The vertices of the bounded region are

(0, 20)

,

(0, 10)

,

(4 + 0.2a, 2 - 0.4a)

, and

(8 + 0.2a, 4 - 0.4a)

.

The region is bounded by the parallel lines

k

and

m

, the perpendicular line

l

, and the

y

-axis (

x = 0

).

4

Find the area of the region as an algebraic expression in terms of

a

.

The area is equal to

60 + 2a

.

The region can be divided into a parallelogram with a vertical base of

10

and width

4 + 0.2a

, and a right triangle with a vertical base of

10

and width

4

. The sum of their areas is

10(4 + 0.2a) + \frac{1}{2}(10)(4) = 40 + 2a + 20 = 60 + 2a

.

5

Set the area expression equal to the given area of

110

and solve for

a

.

a = 25

Setting the area equal to

110

yields

60 + 2a = 110

, which simplifies to

2a = 50

, or

a = 25

.

Key Concept

Linear functions, parallel and perpendicular lines, finding line equations, and coordinate geometry area.

Alternative Method

The area can also be calculated using the geometric properties of a trapezoid. The height of the trapezoid is the perpendicular distance between the parallel lines

k

and

m

, which is

\frac{|20 - 10|}{\sqrt{(-2)^2 + 1^2}} = 2\sqrt{5}

. The bases of the trapezoid are the segments of lines

k

and

m

from the

y

-axis to their intersection points with line

l

. The length of the base on line

k

is

9\sqrt{5}

and the length of the base on line

m

is

13\sqrt{5}

(when

a = 25

). Using the formula for the area of a trapezoid,

\text{Area} = \frac{9\sqrt{5} + 13\sqrt{5}}{2} \times 2\sqrt{5} = 11\sqrt{5} \times 2\sqrt{5} = 110

.

Estimated Time:3m 0s

Question 10Question →

A water tank contains 24 gallons of water. Water is being drained from the tank at a constant rate. After 8 minutes, the tank contains 14 gallons of water. If the volume of water in the tank, in gallons, is a linear function of the time in minutes, how many minutes will it take for the tank to contain exactly 4 gallons of water?

Show answer & explanation

Answer: 16

Answer

16

The volume of water in the tank decreases linearly from an initial value of

24

gallons at

t = 0

to

14

gallons at

t = 8

. The constant rate of change (slope) is calculated by dividing the change in volume by the change in time:

\frac{14 - 24}{8 - 0} = -1.25

gallons per minute. Using the slope-intercept form, the volume

V

at time

t

is given by

V = -1.25t + 24

. Setting the volume

V = 4

gives the equation

4 = -1.25t + 24

. Solving for

t

yields

-20 = -1.25t

, which simplifies to

t = 16

.

Step-by-Step Solution

1

Determine the initial state and the state after 8 minutes as coordinate points.

The initial state is

(0, 24)

and the state after

8

minutes is

(8, 14)

.

These coordinates represent the relationship between time and volume of water in the tank.

2

Calculate the slope (constant rate of change) of the linear function.

Slope

m = \frac{14 - 24}{8 - 0} = -1.25

The slope represents the constant rate at which water is being drained from the tank.

3

Formulate the linear equation.

V(t) = -1.25t + 24

Using the slope-intercept form

V(t) = mt + b

, where

b = 24

is the vertical intercept representing the initial volume.

4

Solve for the time

t

when the volume of water is

4

gallons.

4 = -1.25t + 24 \implies -20 = -1.25t \implies t = 16

To find the time at which the volume decreases to exactly

4

gallons.

Key Concept

Linear Functions and Graphs

Question 11Question →

The table below shows some values of the linear function $f$ .

$x$	$f(x)$
$2$	$11$
$4$	$17$
$6$	$23$

What is the $y$ -coordinate of the $y$ -intercept of the graph of $y = f(x)$ in the $xy$ -plane?

Show answer & explanation

Answer: 5

Answer

The correct answer is 5, representing the y-coordinate of the y-intercept of the graph of f.

To find the

y

-intercept of the linear function, we first determine its slope using the points

(2, 11)

and

(4, 17)

from the table. The slope

m

is

\frac{17 - 11}{4 - 2} = \frac{6}{2} = 3

. Next, we use the slope-intercept equation

f(x) = mx + b

. Substituting

m = 3

and the point

(2, 11)

gives

11 = 3(2) + b

, which simplifies to

11 = 6 + b

. Solving for

b

yields

5

. Therefore, the

y

-coordinate of the

y

-intercept of the graph of

f

is

5

.

Step-by-Step Solution

1

Calculate the slope (

m

) of the linear function using two coordinate pairs from the table.

The slope is

m = 3

.

A linear function has a constant slope, which can be found using the formula

m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}

.

2

Substitute the slope and one of the points into the slope-intercept equation

f(x) = mx + b

to solve for the

y

-intercept

b

.

The

y

-intercept

b

is

5

.

The

y

-coordinate of the

y

-intercept of the graph of

y = f(x)

is the value of

b

in the equation

f(x) = mx + b

.

Key Concept

Finding the y-intercept of a linear function from a table of values.

Estimated Time:45s

Question 12Question →

In the $xy$ -plane, line $l_1$ passes through the origin and has a positive slope. Line $l_2$ is perpendicular to $l_1$ and intersects the $x$ -axis at $(d, 0)$ , where $d > 0$ . The two lines intersect at the point $(a, b)$ in the first quadrant. If the ratio of $a$ to $b$ is $3$ to $4$ , and the area of the triangle bounded by $l_1$ , $l_2$ , and the $x$ -axis is $A$ , which of the following expressions represents $d$ in terms of $A$ ?

\frac{5\sqrt{6A}}{8}

\frac{25A}{6}

\frac{5\sqrt{6A}}{6}

\frac{10\sqrt{6A}}{9}

Show answer & explanation

Answer: $\frac{5\sqrt{6A}}{6}$

Answer

The expression

\frac{5\sqrt{6A}}{6}

represents

d

in terms of

A

.

The correct expression is

\frac{5\sqrt{6A}}{6}

. The slope of line

l_1

passing through the origin and

(a,b)

is given by

m_1 = \frac{b}{a}

. Since the ratio of

a

to

b

is

3

to

4

, we have

\frac{a}{b} = \frac{3}{4}

, which implies the slope

m_1 = \frac{4}{3}

. Because line

l_2

is perpendicular to

l_1

, its slope is

m_2 = -\frac{3}{4}

. The equation of

l_2

passing through

(d,0)

is

y = -\frac{3}{4}(x - d)

. Solving the system of equations gives the intersection coordinates

a = \frac{9}{25}d

and

b = \frac{12}{25}d

. The area of the triangle bounded by the two lines and the

x

-axis is

A = \frac{1}{2} \times d \times b = \frac{6}{25}d^2

. Solving for

d

yields

d = \sqrt{\frac{25A}{6}} = \frac{5\sqrt{6A}}{6}

.

Step-by-Step Solution

1

Determine the slope and equation of line

l_1

.

Slope

m_1 = \frac{4}{3}

, and the equation of the line is

y = \frac{4}{3}x

.

Since

l_1

passes through

(0,0)

and the point

(a,b)

in the first quadrant, its slope is

m_1 = \frac{b}{a}

. Since the ratio of

a

to

b

is

3

to

4

, we have

\frac{a}{b} = \frac{3}{4} \implies \frac{b}{a} = \frac{4}{3}

.

2

Determine the equation of line

l_2

.

The equation of the line is

y = -\frac{3}{4}(x - d)

.

Line

l_2

is perpendicular to

l_1

, so its slope is the negative reciprocal of

m_1

, which is

m_2 = -\frac{3}{4}

. Using the point-slope form with the

x

-intercept

(d,0)

, the equation is

y - 0 = -\frac{3}{4}(x - d)

.

3

Find the coordinates of the intersection point

(a,b)

in terms of

d

.

a = \frac{9}{25}d

and

b = \frac{12}{25}d

.

Set the two equations equal to find the

x

-coordinate of the intersection:

\frac{4}{3}x = -\frac{3}{4}(x - d) \implies \frac{4}{3}x = -\frac{3}{4}x + \frac{3}{4}d \implies \frac{25}{12}x = \frac{3}{4}d \implies x = \frac{9}{25}d

. Substituting this back into the equation of

l_1

yields the

y

-coordinate:

y = \frac{4}{3}\left(\frac{9}{25}d\right) = \frac{12}{25}d

.

4

Express the area of the triangle in terms of

d

.

A = \frac{6}{25}d^2

.

The base of the triangle along the

x

-axis is

d

(from

x=0

to

x=d

). The height of the triangle is the

y

-coordinate of the intersection point,

b = \frac{12}{25}d

. The area of the triangle is given by

A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}d\left(\frac{12}{25}d\right) = \frac{6}{25}d^2

.

5

Solve for

d

in terms of

A

.

d = \frac{5\sqrt{6A}}{6}

.

Isolate

d^2

to get

d^2 = \frac{25A}{6}

. Taking the square root of both sides gives

d = \sqrt{\frac{25A}{6}} = \frac{5\sqrt{A}}{\sqrt{6}} = \frac{5\sqrt{6A}}{6}

.

Key Concept

Using the properties of perpendicular lines, setting up equations from coordinate parameters, and applying geometric formulas to relate parameters in linear systems.

Alternative Method

Using the geometric mean theorem (altitude rule) in a right triangle, the altitude

b

divides the hypotenuse

d

into segments

a

and

d-a

, such that

b^2 = a(d-a)

. Since

b = \frac{4}{3}a

, we can substitute this to find

a = \frac{9}{25}d

and

b = \frac{12}{25}d

directly without finding the line equations. Then,

A = \frac{1}{2} d b = \frac{6}{25}d^2

, which solves to

d = \frac{5\sqrt{6A}}{6}

.

Estimated Time:3m 0s

Question 13Question →

In the $xy$ -plane, the graph of the linear function $f$ is perpendicular to the line with equation $3x + 4y = 48$ . The graph of $f$ intersects the $x$ -axis at the point $(p, 0)$ and the $y$ -axis at the point $(0, q)$ , where $q > 0$ . If the distance between the points $(p, 0)$ and $(0, q)$ is $15$ , what is the value of $q$ ?

Show answer & explanation

Answer: 12

Answer

12

First, find the slope of the given line by rewriting

3x + 4y = 48

in slope-intercept form:

y = -\frac{3}{4}x + 12

. The slope is

-\frac{3}{4}

. The graph of the linear function

f

is perpendicular to this line, so its slope is the negative reciprocal,

\frac{4}{3}

. With a

y

-intercept of

(0, q)

, the equation of

f

is

y = \frac{4}{3}x + q

. Setting

y = 0

gives the

x

-intercept

(p, 0) = (-\frac{3}{4}q, 0)

. The distance between these intercepts is

\sqrt{(-\frac{3}{4}q)^2 + q^2} = \sqrt{\frac{25}{16}q^2} = \frac{5}{4}q

(since

q > 0

). Given that the distance is

15

, we solve

\frac{5}{4}q = 15

to find

q = 12

.

Step-by-Step Solution

1

Find the slope of the line

3x + 4y = 48

.

The slope of the line is

-\frac{3}{4}

.

To find the slope of the perpendicular line

f

, we first need the slope of the given line. Rewriting

3x + 4y = 48

in slope-intercept form gives

y = -\frac{3}{4}x + 12

.

2

Determine the slope of

f

.

The slope of

f

is

\frac{4}{3}

.

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Find the

x

-intercept of

f

in terms of

q

.

The

x

-intercept of

f

is

(-\frac{3}{4}q, 0)

.

Since the

y

-intercept of

f

is

(0, q)

, the equation of

f

is

y = \frac{4}{3}x + q

. Setting

y = 0

gives

0 = \frac{4}{3}x + q

, which simplifies to

x = -\frac{3}{4}q

.

4

Use the distance formula between the intercepts to solve for

q

.

The value of

q

is

12

.

The distance between

(-\frac{3}{4}q, 0)

and

(0, q)

is

\sqrt{(-\frac{3}{4}q)^2 + q^2} = \sqrt{\frac{25}{16}q^2} = \frac{5}{4}q

since

q > 0

. Setting this distance to

15

gives

\frac{5}{4}q = 15

, which yields

q = 12

.

Key Concept

Properties of perpendicular lines, finding intercepts, and utilizing the distance formula in coordinate geometry.

Question 14Question →

In the $xy$ -plane, the graph of a linear function $f$ has a $y$ -intercept of $(0, r)$ and an $x$ -intercept of $(s, 0)$ , where $r$ and $s$ are nonzero constants. A second line, $g$ , is perpendicular to the graph of $f$ and passes through the point $(s, r)$ . Which of the following expressions represents the $x$ -intercept of line $g$ ?

\frac{s^2 - r^2}{r}

\frac{s^2 + r^2}{s}

\frac{s^2 - r^2}{s}

\frac{r^2 - s^2}{r}

Show answer & explanation

Answer: $\frac{s^2 - r^2}{s}$

Answer

The expression

\frac{s^2 - r^2}{s}

The slope of line

f

is found using its intercepts

(0, r)

and

(s, 0)

, which gives a slope of

m_f = -\frac{r}{s}

. Since line

g

is perpendicular to line

f

, its slope is the negative reciprocal,

m_g = \frac{s}{r}

. Using the point-slope form with the point

(s, r)

, the equation for line

g

is

y - r = \frac{s}{r}(x - s)

. Setting

y = 0

to find the

x

-intercept yields

x = \frac{s^2 - r^2}{s}

.

Step-by-Step Solution

1

Determine the slope of the linear function

f

using its intercepts.

The slope of

f

is

m_f = \frac{0 - r}{s - 0} = -\frac{r}{s}

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

m = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Find the slope of line

g

, which is perpendicular to the graph of

f

.

The slope of line

g

is

m_g = -\frac{1}{m_f} = -\frac{1}{-r/s} = \frac{s}{r}

.

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Write the equation of line

g

in point-slope form using the point

(s, r)

.

The equation of line

g

is

y - r = \frac{s}{r}(x - s)

.

The point-slope form of a linear equation is

y - y_1 = m(x - x_1)

.

4

Solve for the

x

-intercept of line

g

by setting

y = 0

.

Setting

y = 0

gives

-r = \frac{s}{r}(x - s) \implies -\frac{r^2}{s} = x - s \implies x = s - \frac{r^2}{s} = \frac{s^2 - r^2}{s}

.

The

x

-intercept of a line is the value of

x

when

y = 0

.

Key Concept

Finding the equation of a perpendicular line and its intercepts using algebraic representations.

Question 15Question →

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(-2, c)$ and $(c, 7)$ , where $c$ is a constant. If the slope of the graph of $f$ is $\frac{1}{2}$ , what is the value of $c$ ?

1

4

6

16

Show answer & explanation

Answer: 4

Answer

The value of

c

is 4.

Using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

with the points

(-2, c)

and

(c, 7)

and the slope

m = \frac{1}{2}

gives

\frac{7 - c}{c - (-2)} = \frac{1}{2}

. Simplifying the denominator yields

\frac{7 - c}{c + 2} = \frac{1}{2}

. Cross-multiplying gives

2(7 - c) = c + 2

, which expands to

14 - 2c = c + 2

. Adding

2c

to both sides and subtracting

2

from both sides gives

12 = 3c

. Dividing both sides by

3

gives

c = 4

.

Step-by-Step Solution

1

Set up the slope formula for the line passing through

(-2, c)

and

(c, 7)

.

m = \frac{7 - c}{c - (-2)}

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

m = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Substitute the given slope

m = \frac{1}{2}

into the formula and simplify the denominator.

\frac{7 - c}{c + 2} = \frac{1}{2}

We are given that the slope of the line is

\frac{1}{2}

.

3

Solve the equation for

c

by cross-multiplying and isolating the variable.

2(7 - c) = c + 2 \implies 14 - 2c = c + 2 \implies 12 = 3c \implies c = 4

Cross-multiplying eliminates the fractions, allowing us to solve the linear equation for

c

.

Key Concept

Finding the slope of a line given two points containing an unknown coordinate

Question 16Question →

In the $xy$ -plane, the graph of a linear function $f$ has a positive $y$ -intercept and a positive $x$ -intercept. The area of the triangular region in the first quadrant bounded by the graph of $f$ and the coordinate axes is $36$ . If the graph of $f$ passes through the point $(4, 4)$ and has a slope less than $-1$ , what is the $y$ -coordinate of the $y$ -intercept of the graph of $f$ ?

Show answer & explanation

Answer: 12

Answer

The

y

-coordinate of the

y

-intercept of the graph of

f

is

12

.

The correct answer is

12

because the system of equations derived from the area constraint (

qr = 72

) and the point constraint (

q+r=18

) yields two possible values for the

y

-intercept:

6

or

12

. The condition that the slope must be less than

-1

means that the line must be steeper than a slope of

-1

, which requires the

y

-intercept to be larger than the

x

-intercept (

r > q

). Thus, the

y

-intercept is

12

.

Step-by-Step Solution

1

Express the equation of the line using intercept form.

\frac{x}{q} + \frac{y}{r} = 1

, where

q > 0

is the

x

-intercept and

r > 0

is the

y

-intercept.

Since the line intersects the positive axes, this form directly relates the intercepts to the coordinates of points on the line.

2

Use the area of the triangle to find a relationship between

q

and

r

.

qr = 72

The area of the right triangle formed by the axes and the intercepts is given by

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}qr = 36

.

3

Substitute the point

(4, 4)

into the equation of the line and simplify using the area relationship.

q + r = 18

Plugging in

x = 4

and

y = 4

yields

\frac{4}{q} + \frac{4}{r} = 1 \implies 4(q + r) = qr

. Substituting

qr = 72

gives

4(q + r) = 72 \implies q + r = 18

.

4

Solve the system of equations

q + r = 18

and

qr = 72

.

(q, r) = (6, 12)

or

(q, r) = (12, 6)

Substituting

r = 18 - q

into

qr = 72

yields

q(18 - q) = 72 \implies q^2 - 18q + 72 = 0 \implies (q - 6)(q - 12) = 0

.

5

Apply the slope condition to determine the unique value of

r

.

r = 12

The slope of the line is

m = -\frac{r}{q}

. If

q = 12

and

r = 6

, then

m = -0.5

, which is not less than

-1

. If

q = 6

and

r = 12

, then

m = -2

, which is less than

-1

. Thus,

r = 12

is the correct

y

-intercept.

Key Concept

Using intercepts and area to determine the equation of a linear function under constraints.

Question 17Question →

The table below shows some values of $x$ and the corresponding values of $f(x)$ for a linear function $f$ .

$x$	$f(x)$
3	7
6	12
9	17

What is the value of $f(15)$ ?

Show answer & explanation

Answer: 27

Answer

The value of the linear function evaluated at 15 is 27.

To find

f(15)

for the linear function, we determine its constant rate of change. Using points

(3, 7)

and

(6, 12)

, the change in

f(x)

is

12 - 7 = 5

for a change in

x

of

6 - 3 = 3

. This gives a slope of

\frac{5}{3}

. Using point-slope form with

(3, 7)

gives the equation

f(x) - 7 = \frac{5}{3}(x - 3)

, which simplifies to

f(x) = \frac{5}{3}x + 2

. Substituting

x = 15

yields

f(15) = \frac{5}{3}(15) + 2 = 25 + 2 = 27

. Alternatively, since

x

increases by 6 from 9 to 15, which is twice the step size of 3, the value of

f(x)

must increase by

2 \times 5 = 10

from

17

, resulting in

17 + 10 = 27

.

Step-by-Step Solution

1

Calculate the slope of the linear function

slope

m = \frac{5}{3}

Linear functions have a constant rate of change, which is the slope.

2

Determine the equation of the function

f(x) = \frac{5}{3}x + 2

Using the slope and one point from the table helps define the function for all inputs.

3

Evaluate the function at

x = 15

f(15) = 27

Substitute the given input value into the function equation to find the corresponding output value.

Key Concept

Linear Functions and Graphs

Question 18Question →

A line in the $xy$ -plane has the equation $3y - 2x = 12$ . What is the slope of this line?

\frac{3}{2}

4

\frac{2}{3}

-\frac{2}{3}

Show answer & explanation

Answer: $\frac{2}{3}$

Answer

The slope of the line is

\frac{2}{3}

.

To find the slope, the equation

3y - 2x = 12

is rewritten in slope-intercept form (

y = mx + b

). Adding

2x

to both sides gives

3y = 2x + 12

. Dividing by

3

yields

y = \frac{2}{3}x + 4

. The slope is the coefficient of

x

, which is

\frac{2}{3}

.

Step-by-Step Solution

1

Isolate the term containing

y

by adding

2x

to both sides of the equation.

3y = 2x + 12

To express the equation in slope-intercept form (

y = mx + b

).

2

Divide all terms in the equation by

3

to solve for

y

.

y = \frac{2}{3}x + 4

To isolate

y

and identify the slope,

m

, which is the coefficient of

x

.

Key Concept

Finding the slope of a line from its linear equation by rewriting it in slope-intercept form.

Question 19Question →

A linear function $f$ has a slope of $\frac{2}{3}$ . The function $g$ is defined by $g(x) = f(x + 4) - 3$ . If the graph of $y = g(x)$ in the $xy$ -plane has an $x$ -intercept at $(5, 0)$ , what is the $x$ -intercept of the graph of $y = f(x)$ ?

Show answer & explanation

Answer: 4.5

Answer

4.5

The correct answer is

4.5

(or

\frac{9}{2}

). By using the function transformation relation, we find

g(5) = f(9) - 3 = 0

, which yields

f(9) = 3

. With a slope of

\frac{2}{3}

, the equation of the line is

f(x) = \frac{2}{3}x - 3

. Setting

f(x) = 0

yields the

x

-intercept at

x = 4.5

.

Step-by-Step Solution

1

Use the definition of function

g

and its given

x

-intercept to find a point on the graph of

f

.

f(9) = 3

Since the graph of

y = g(x)

has an

x

-intercept at

(5, 0)

, we know

g(5) = 0

. Substituting

x = 5

into the definition

g(x) = f(x + 4) - 3

gives

g(5) = f(5 + 4) - 3 = f(9) - 3

. Since

g(5) = 0

, it follows that

f(9) - 3 = 0

, or

f(9) = 3

.

2

Determine the equation of the linear function

f

using its slope and the point identified in Step 1.

f(x) = \frac{2}{3}x - 3

The function

f

is linear with a slope of

\frac{2}{3}

and passes through the point

(9, 3)

. Using the point-slope formula, we get

f(x) - 3 = \frac{2}{3}(x - 9)

, which simplifies to

f(x) = \frac{2}{3}x - 3

.

3

Find the

x

-intercept of the graph of

y = f(x)

by setting

f(x) = 0

.

x = 4.5

(or

\frac{9}{2}

)

To find the

x

-intercept, set

f(x) = 0

. This gives

\frac{2}{3}x - 3 = 0

. Adding

3

to both sides and multiplying by

\frac{3}{2}

yields

x = \frac{9}{2}

, which is equal to

4.5

.

Key Concept

Linear Functions and Graphs

Alternative Method

Alternatively, we can write the equation of

f

in slope-intercept form as

f(x) = \frac{2}{3}x + b

. Then the definition of

g(x)

becomes

g(x) = \frac{2}{3}(x + 4) + b - 3 = \frac{2}{3}x + \frac{8}{3} + b - 3 = \frac{2}{3}x + b - \frac{1}{3}

. Since the graph of

g

has an

x

-intercept at

(5, 0)

, we substitute

x = 5

and

g(5) = 0

to get

\frac{2}{3}(5) + b - \frac{1}{3} = 0

, which simplifies to

3 + b = 0

, so

b = -3

. This gives

f(x) = \frac{2}{3}x - 3

. Finally, the

x

-intercept is found by solving

\frac{2}{3}x - 3 = 0

, resulting in

x = 4.5

.

Estimated Time:2m 0s

Question 20Question →

In the $xy$ -plane, line $L$ passes through the points $(0, 4)$ and $(6, 0)$ . A second line, $M$ , is perpendicular to line $L$ and intersects line $L$ at a point on the line $y = x$ . If line $M$ is represented by the equation $y = px + q$ , where $p$ and $q$ are constants, what is the value of $p + q$ ?

\frac{6}{5}

\frac{10}{3}

\frac{3}{10}

-\frac{9}{2}

Show answer & explanation

Answer: $\frac{3}{10}$

Answer

The correct answer is the value stating three-tenths

To find the sum of the slope

p

and

y

-intercept

q

of line

M

, we first determine the equation of line

L

. Line

L

passes through

(0, 4)

and

(6, 0)

, so its slope is

\frac{0 - 4}{6 - 0} = -\frac{2}{3}

. The equation of line

L

is therefore

y = -\frac{2}{3}x + 4

. Setting

y = x

gives

x = -\frac{2}{3}x + 4

, which yields

x = 2.4

. The intersection point is

(2.4, 2.4)

. Because line

M

is perpendicular to line

L

, its slope is the negative reciprocal of

-\frac{2}{3}

, which is

p = 1.5

. Using point-slope form with the intersection point

(2.4, 2.4)

, the equation of line

M

is

y - 2.4 = 1.5(x - 2.4)

, which simplifies to

y = 1.5x - 1.2

. Thus,

q = -1.2

. The sum

p + q

is

1.5 - 1.2 = 0.3

, or

\frac{3}{10}

.

Step-by-Step Solution

1

Find the equation of line

L

using the given points

(0, 4)

and

(6, 0)

.

The slope of line

L

is

m_L = \frac{0 - 4}{6 - 0} = -\frac{2}{3}

. Since the

y

-intercept is

(0, 4)

, the equation of line

L

is

y = -\frac{2}{3}x + 4

.

To find where line

L

intersects another line, we first need to establish its linear equation.

2

Find the intersection point of line

L

and the line

y = x

.

Substitute

y = x

into the equation for line

L

:

x = -\frac{2}{3}x + 4 \implies \frac{5}{3}x = 4 \implies x = 2.4

. Thus, the intersection point is

(2.4, 2.4)

.

The problem states that line

M

intersects line

L

at a point on the line

y = x

.

3

Determine the equation of line

M

using the intersection point and the perpendicular relationship.

Since line

M

is perpendicular to line

L

, its slope

p

is the negative reciprocal of

-\frac{2}{3}

, which is

p = \frac{3}{2} = 1.5

. Using point-slope form at

(2.4, 2.4)

:

y - 2.4 = 1.5(x - 2.4) \implies y = 1.5x - 1.2

. This gives

q = -1.2

.

Perpendicular lines in the coordinate plane have slopes that multiply to negative one. We then use the point-slope formula to define the equation of line

M

.

4

Calculate the sum of

p

and

q

.

p + q = 1.5 + (-1.2) = 0.3 = \frac{3}{10}

.

The question asks for the value of the expression

p + q

.

Key Concept

Perpendicular lines and linear graph intersections

Estimated Time:3m 0s

Linear Functions and Graphs

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Common Mistakes

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Common Mistakes

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Alternative Method

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

Alternative Method

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Alternative Method

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes