Linear Functions and Graphs Practice Questions — SAT — Page 4

Question 61Question →

In the $xy$ -plane, the graph of a line $l$ passes through the points $(0, 1)$ and $(3, 5)$ . If another point on line $l$ has coordinates $(t, 9)$ , what is the value of $t$ ?

Show answer & explanation

Answer: 6

Answer

The value of

t

is

6

.

The slope of line

l

is

m = \frac{5 - 1}{3 - 0} = \frac{4}{3}

. Using the y-intercept

(0, 1)

, the equation of the line is

y = \frac{4}{3}x + 1

. Setting

y = 9

gives

9 = \frac{4}{3}t + 1

. Subtracting 1 from both sides gives

8 = \frac{4}{3}t

. Multiplying both sides by

\frac{3}{4}

yields

t = 6

.

Step-by-Step Solution

1

Calculate the slope of line

l

using the slope formula

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

with the points

(0, 1)

and

(3, 5)

.

m = \frac{5 - 1}{3 - 0} = \frac{4}{3}

The slope of a line represents its constant rate of change and is needed to determine the line's equation.

2

Write the equation of the line in slope-intercept form,

y = mx + b

, using the slope

m = \frac{4}{3}

and the y-intercept

b = 1

(from the point

(0, 1)

).

y = \frac{4}{3}x + 1

The slope-intercept equation defines the relationship between the

x

- and

y

-coordinates of any point on the line.

3

Substitute the point

(t, 9)

into the line's equation and solve for

t

.

9 = \frac{4}{3}t + 1 \implies 8 = \frac{4}{3}t \implies t = 6

Since the point lies on the line, its coordinates must satisfy the line's equation.

Key Concept

Determining the equation of a linear function from a graph or points and evaluating it for a given value.

Question 62Question →

An online retailer uses a linear relationship to determine the shipping cost of an order based on the total weight of the items in the order. The table below shows the shipping cost, $y$ , in dollars, for an order with a total weight of $x$ pounds.

Weight ( $x$ pounds)	Shipping Cost ( $y$ dollars)
$2$	$9.50$
$6$	$17.50$

What is the shipping cost, in dollars, for an order with a total weight of $11$ pounds?

Show answer & explanation

Answer: 27.5

Answer

The correct shipping cost for an order weighing 11 pounds is 27.5 dollars.

The linear relationship has a constant rate of change (slope). Using the coordinates from the table,

(2, 9.50)

and

(6, 17.50)

, the slope

m

is calculated as

\frac{17.50 - 9.50}{6 - 2} = \frac{8}{4} = 2

. Using the point-slope form, the equation of the line is

y - 9.50 = 2(x - 2)

, which simplifies to

y = 2x + 5.50

. Substituting

x = 11

gives

y = 2(11) + 5.50 = 27.5

.

Step-by-Step Solution

1

Calculate the slope (rate of change) of the linear relationship.

m = 2

Determines the rate at which the shipping cost increases per additional pound of weight.

2

Formulate the linear equation representing the relationship.

y = 2x + 5.50

Establishes a functional relationship to calculate costs for any given weight.

3

Evaluate the function at the target weight of 11 pounds.

y = 27.5

Determines the final cost of shipping for the specified weight of 11 pounds.

Key Concept

Determining and evaluating linear functions represented in tabular form

Estimated Time:1m 30s

Question 63Question →

Company A and Company B both rent moving trucks. The total cost $A(d)$ , in dollars, for renting a truck from Company A for $d$ days is given by $A(d) = 35d + 75$ . The table below shows some values of the total cost $B(d)$ , in dollars, for renting a truck from Company B for $d$ days.

Days ( $d$ )	Total Cost ( $B(d)$ )
2	150
5	300

If the relationship between $d$ and $B(d)$ is linear, for how many days of rental will the total cost at Company B be exactly $95$ dollars more than the total cost at Company A?

2

3

5

8

Show answer & explanation

Answer: 8

Answer

The total cost at Company B will be exactly 95 dollars more than the total cost at Company A for a rental of 8 days.

To find the number of days for which Company B's cost is exactly 95 dollars more than Company A's cost, we first find the linear function for Company B. The rate of change is

\frac{300 - 150}{5 - 2} = 50

dollars per day. The initial cost is

150 - 50(2) = 50

dollars, giving

B(d) = 50d + 50

. We then set up the equation

B(d) - A(d) = 95

, which is

(50d + 50) - (35d + 75) = 95

. Simplifying this equation gives

15d - 25 = 95

. Adding 25 to both sides gives

15d = 120

, and dividing by 15 gives

d = 8

.

Step-by-Step Solution

1

Determine the linear cost function for Company B,

B(d) = md + b

, using the points

(2, 150)

and

(5, 300)

from the table.

The slope is

m = \frac{300 - 150}{5 - 2} = 50

. Using

(2, 150)

to find the y-intercept:

150 = 50(2) + b \Rightarrow b = 50

. Thus,

B(d) = 50d + 50

.

We need to find the equation representing the total cost of Company B to set up the comparison.

2

Set up an equation where the total cost at Company B is equal to the total cost at Company A plus 95 dollars.

B(d) = A(d) + 95 \Rightarrow 50d + 50 = (35d + 75) + 95

.

This represents the condition that Company B's cost is 95 dollars more than Company A's cost.

3

Simplify the equation and solve for the number of days,

d

.

50d + 50 = 35d + 170 \Rightarrow 15d = 120 \Rightarrow d = 8

.

Isolating the variable

d

gives the exact number of days required.

Key Concept

Comparing linear functions and solving linear equations representing real-world contexts.

Question 64Question →

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(3, 11)$ and $(7, 23)$ . The function $g$ is defined by $g(x) = f(2x) - 5$ . What is the value of $g(4)$ ?

Show answer & explanation

Answer: 21

Answer

21

The correct answer is 21. First, find the slope of the linear function

f

using the given points

(3, 11)

and

(7, 23)

:

m = \frac{23 - 11}{7 - 3} = 3

. The equation of the line is

f(x) - 11 = 3(x - 3)

, which simplifies to

f(x) = 3x + 2

. To find

g(4)

, substitute

x = 4

into the definition of

g(x)

:

g(4) = f(2 \cdot 4) - 5 = f(8) - 5

. Evaluating

f(8)

gives

3(8) + 2 = 26

. Finally, subtracting

5

gives

g(4) = 26 - 5 = 21

.

Step-by-Step Solution

1

Determine the equation of the linear function

f(x)

f(x) = 3x + 2

First find the slope

m = \frac{23 - 11}{7 - 3} = 3

. Then, use the point-slope formula with

(3, 11)

to get

f(x) - 11 = 3(x - 3)

, which simplifies to

f(x) = 3x + 2

.

2

Express

g(4)

in terms of

f

g(4) = f(8) - 5

Substitute

x = 4

into the definition

g(x) = f(2x) - 5

to get

g(4) = f(2(4)) - 5

.

3

Calculate the value of

f(8)

and

g(4)

g(4) = 21

Evaluate

f(8) = 3(8) + 2 = 26

, then subtract

5

to obtain

g(4) = 26 - 5 = 21

.

Key Concept

Linear Functions and Graphs

Question 65Question →

A scientist is measuring the temperature of a chemical compound during an experiment. The temperature of the compound decreases at a constant rate. At $t = 4$ minutes after the experiment begins, the temperature of the compound is $72^\circ\text{C}$ . At $t = 12$ minutes, the temperature of the compound is $48^\circ\text{C}$ . Which of the following functions $T$ models the temperature, in degrees Celsius, of the compound $t$ minutes after the experiment begins, where $0 \leq t \leq 25$ ?

T(t) = -3t + 84

T(t) = -3t + 72

T(t) = -\frac{1}{3}t + \frac{220}{3}

T(t) = -3t + 60

Show answer & explanation

Answer: $T(t) = -3t + 84$

Answer

T(t) = -3t + 84

To find the linear model

T(t) = mt + b

, we first calculate the slope

m

using the given points

(4, 72)

and

(12, 48)

. The slope is the change in temperature divided by the change in time:

m = \frac{48 - 72}{12 - 4} = -3

. Next, we find the y-intercept

b

by substituting one of the points into the equation

T(t) = mt + b

. Substituting

t = 4

and

T(4) = 72

yields

72 = -3(4) + b

, which simplifies to

72 = -12 + b

. Adding

12

to both sides gives

b = 84

. Therefore, the correct linear function modeling the temperature is

T(t) = -3t + 84

.

Step-by-Step Solution

1

Calculate the rate of temperature change (slope) per minute.

The slope

m

is

-3

.

Since the temperature decreases at a constant rate, the relationship is linear. The slope

m

is the change in temperature divided by the change in time:

m = \frac{48 - 72}{12 - 4} = \frac{-24}{8} = -3

.

2

Find the initial temperature (y-intercept) at

t = 0

.

The y-intercept

b

is

84

.

Substitute the point

(4, 72)

and the slope

m = -3

into the slope-intercept equation

T(t) = mt + b

. This gives

72 = -3(4) + b

, which simplifies to

72 = -12 + b

. Adding

12

to both sides yields

b = 84

.

3

Write the final equation.

T(t) = -3t + 84

Combine the slope of

-3

and the y-intercept of

84

to complete the linear model equation.

Key Concept

Writing and interpreting linear functions from two points

Alternative Method

Instead of deriving the function algebraically, substitute the given values of

t = 4

and

t = 12

into each of the given options. Only the correct option yields the corresponding temperatures of 72 and 48 degrees Celsius, respectively.

Estimated Time:1m 30s

Question 66Question →

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

$x$	$f(x)$
$2$	$5$
$5$	$14$
$8$	$23$

If the function $g$ is defined by $g(x) = f(x) + k$ , where $k$ is a constant, and the $x$ -intercept of the graph of $y = g(x)$ in the $xy$ -plane is $-2$ , what is the value of $k$ ?

Show answer & explanation

Answer: 7

Answer

The value of

k

is

7

.

To find the value of

k

, we first determine the linear function

f(x)

using the values in the table. The slope of

f

is calculated as

\frac{14 - 5}{5 - 2} = 3

. Substituting the point

(2, 5)

into the slope-intercept form

f(x) = 3x + b

gives

5 = 3(2) + b

, which simplifies to

b = -1

. Therefore,

f(x) = 3x - 1

. The function

g

is defined as

g(x) = f(x) + k = 3x - 1 + k

. Since the

x

-intercept of the graph of

y = g(x)

is

-2

, we must have

g(-2) = 0

. Substituting

x = -2

into the equation for

g(x)

gives

3(-2) - 1 + k = 0

, which simplifies to

-7 + k = 0

. Solving for

k

yields

k = 7

.

Step-by-Step Solution

1

Determine the slope of the linear function

f

.

The slope of

f

is

3

.

Using the points

(2, 5)

and

(5, 14)

from the table, the slope is calculated as

m = \frac{14 - 5}{5 - 2} = \frac{9}{3} = 3

.

2

Find the equation for

f(x)

.

f(x) = 3x - 1

Using the slope-intercept form

f(x) = 3x + b

and the point

(2, 5)

, we solve for

b

:

5 = 3(2) + b

, which gives

b = -1

.

3

Set up the equation for

g(x)

and use the

x

-intercept to find the value of

k

.

k = 7

Since

g(x) = f(x) + k

, we have

g(x) = 3x - 1 + k

. The

x

-intercept of

g(x)

is

-2

, so

g(-2) = 0

. This gives

3(-2) - 1 + k = 0

, which simplifies to

-7 + k = 0

, meaning

k = 7

.

Key Concept

Finding the equation of a linear function from tabular data and applying translations and intercepts.

Question 67Question →

An agricultural drone is spraying liquid fertilizer on a field at a constant rate. The total amount of fertilizer remaining in the drone's tank, in liters, can be modeled by a linear function of the time, in minutes, since the drone began spraying. After $3$ minutes of spraying, $84$ liters of fertilizer remain in the tank. After $8$ minutes of spraying, $54$ liters of fertilizer remain in the tank. How many minutes of spraying will it take for the tank to become completely empty?

Show answer & explanation

Answer: 17

Answer

17

The correct answer is 17. The volume of fertilizer in the tank decreases at a constant rate of 6 liters per minute. Since 84 liters remain after 3 minutes, the initial volume of fertilizer in the tank was 102 liters. Dividing the initial volume of 102 liters by the rate of 6 liters per minute yields 17 minutes for the tank to be completely empty.

Step-by-Step Solution

1

Calculate the constant rate of change (slope) of the remaining fertilizer volume.

-6 liters per minute

The slope formula is used with the two data points representing time and volume:

(3, 84)

and

(8, 54)

.

2

Determine the initial volume of fertilizer in the tank (y-intercept).

102 liters

Using the slope-intercept form

F(t) = mt + b

, we substitute the slope

m = -6

and the point

(3, 84)

to solve for

b

.

3

Find the time when the volume of remaining fertilizer reaches 0.

17 minutes

Setting the linear function equal to 0 and solving for time gives the total duration until the tank is empty.

Key Concept

Linear Functions and Graphs

Question 68Question →

In the $xy$ -plane, the graph of a linear function $f$ passes through the points $(k, 2k + 3)$ and $(2k, 5k - 1)$ , where $k$ is a constant. If the $y$ -intercept of the graph of $f$ is $-9$ , what is the value of $k$ ?

Show answer & explanation

Answer: 16

Answer

The value of

k

is 16.

The value of

k

is 16 because when

k = 16

, the points on the graph are

(16, 35)

and

(32, 79)

. The slope of the line is

\frac{79 - 35}{32 - 16} = 2.75

. The equation of the line in slope-intercept form is

y = 2.75x + b

. Using the point

(16, 35)

, we get

35 = 2.75(16) + b \implies 35 = 44 + b \implies b = -9

, which matches the given

y

-intercept of

-9

.

Step-by-Step Solution

1

Calculate the slope of the line in terms of

k

using the two given points

(k, 2k + 3)

and

(2k, 5k - 1)

.

The slope

m

is

\frac{3k - 4}{k}

.

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

Use the

y

-intercept of

-9

, which corresponds to the point

(0, -9)

, along with the point

(k, 2k + 3)

to write another expression for the slope.

The slope

m

is

\frac{2k + 12}{k}

.

The slope must be constant for all points on the line, so the slope between the

y

-intercept and one of the points must equal the slope between the two points.

3

Equate the two slope expressions and solve for

k

.

k = 16

Setting the two expressions for the slope equal to each other gives

\frac{3k - 4}{k} = \frac{2k + 12}{k}

. Multiplying by

k

on both sides yields

3k - 4 = 2k + 12

, which simplifies to

k = 16

.

Key Concept

Linear Functions and Graphs

Question 69Question →

A line in the $xy$ -plane representing the linear function $f$ has a $y$ -intercept of $(0, 8)$ and contains the point $(a, b)$ , where $a$ and $b$ are positive integers. If the slope of this line is $-\frac{3}{5}$ and the sum of $a$ and $b$ is $12$ , what is the value of $a$ ?

2

6

8

10

Show answer & explanation

Answer: 10

Answer

10

Since the line representing the function has a y-intercept of (0, 8) and a slope of -3/5, its equation in slope-intercept form is y = -3/5x + 8. Since the line contains the point (a, b), substituting x = a and y = b into this equation gives b = -3/5a + 8. The sum of a and b is given as 12, so a + b = 12, which can be rewritten as b = 12 - a. Substituting this expression for b gives the equation 12 - a = -3/5a + 8. Subtracting 8 from both sides yields 4 - a = -3/5a. Adding a to both sides yields 4 = 2/5a. Multiplying both sides by 5/2 gives a = 10.

Step-by-Step Solution

1

Write the equation of the linear function in slope-intercept form.

y = -\frac{3}{5}x + 8

The y-intercept of the line is given as (0, 8), and the slope of the line is given as -3/5.

2

Substitute the point (a, b) into the equation of the line.

b = -\frac{3}{5}a + 8

Since the point (a, b) lies on the line, its coordinates must satisfy the equation of the line.

3

Express b in terms of a using the given sum relationship.

b = 12 - a

We are given that the sum of a and b is 12, which can be written as a + b = 12.

4

Substitute the expression for b into the line's equation and solve for a.

a = 10

Substituting b = 12 - a into b = -3/5a + 8 yields 12 - a = -3/5a + 8, which simplifies to 4 = 2/5a, leading to a = 10.

Key Concept

Linear Functions and Graphs

Question 70Question →

A linear function $f$ is defined such that its graph passes through the points $(2, 5)$ and $(4, 11)$ . The graph of another linear function, $g$ , is perpendicular to the graph of $f$ . If the graph of $g$ passes through the point $(3, 10)$ , what is the $x$ -coordinate of the $x$ -intercept of the graph of $g$ ?

Show answer & explanation

Answer: 33

Answer

The correct answer is

33

.

The slope of the linear function

f

is found using the two given points:

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

. Because the graph of the linear function

g

is perpendicular to the graph of

f

, its slope is the negative reciprocal of

3

, which is

-\frac{1}{3}

. Using the point-slope form with the point

(3, 10)

, the equation of

g

is

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

, which simplifies to

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

. The

x

-intercept is the value of

x

when

y = 0

. Setting

y

to

0

gives

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

, which yields

x = 33

.

Step-by-Step Solution

1

Calculate the slope of the linear function

f

The slope of

f

is

3

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

. Substituting

(2, 5)

and

(4, 11)

yields

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

.

2

Determine the slope of the perpendicular linear function

g

The slope of

g

is

-\frac{1}{3}

.

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

3

is

-\frac{1}{3}

.

3

Write the equation representing function

g

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

Using the point-slope equation

y - y_1 = m(x - x_1)

with the slope

m = -\frac{1}{3}

and the point

(3, 10)

, we get

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

. Distributing and simplifying yields

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

.

4

Find the

x

-intercept of the graph of

g

33

The

x

-intercept is the point on the graph where

y = 0

. Substituting

0

for

y

in the equation of

g

gives

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

.

Key Concept

The relationship between the slopes of perpendicular lines and finding intercepts of linear functions.

Estimated Time:1m 30s

Question 71Question →

A construction company rents a concrete mixer. The total cost, in dollars, of renting the concrete mixer is a linear function of the rental time, in hours. The table below shows some values of the rental time and the corresponding total cost.

Time (hours)	Total cost (dollars)
$3$	$54$
$5$	$78$
$8$	$114$

Which of the following is the total cost, in dollars, of renting the concrete mixer for $12$ hours?

55

144

162

168

Show answer & explanation

Answer: $162$

Answer

162

dollars

The cost function is linear, represented by

C(t) = mt + b

. The rate of change (slope) is calculated as

m = \frac{78 - 54}{5 - 3} = 12

dollars per hour. Using the point

(3, 54)

, we solve for the initial fee (y-intercept)

b

:

54 = 12(3) + b

, which gives

b = 18

. Thus, the linear model is

C(t) = 12t + 18

. For a rental time of

12

hours, the total cost is

C(12) = 12(12) + 18 = 162

dollars.

Step-by-Step Solution

1

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

m = 12

Since the relationship is linear, the rate of change is constant. Using the coordinate points

(3, 54)

and

(5, 78)

from the table, the slope is calculated as

m = \frac{78 - 54}{5 - 3} = 12

.

2

Determine the y-intercept (initial fee) of the function.

b = 18

Substitute one of the points, such as

(3, 54)

, and the slope

m = 12

into the slope-intercept form equation

C(t) = mt + b

. This gives

54 = 12(3) + b \Rightarrow 54 = 36 + b \Rightarrow b = 18

.

3

Evaluate the cost function for a rental time of

12

hours.

C(12) = 162

Using the complete linear equation

C(t) = 12t + 18

, substitute

t = 12

to find the total cost:

C(12) = 12(12) + 18 = 144 + 18 = 162

.

Key Concept

Finding and evaluating a linear function from tabular data

Estimated Time:1m 30s

Linear Functions and Graphs

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes