Linear Functions and Graphs Alıştırma Soruları — SAT — Sayfa 3

Soru 41Soru →

In the $xy$ -plane, the graph of the linear function $h$ has a slope of $3$ and passes through the point $(5, 8)$ . What is the value of $h(11)$ ?

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Cevap: 26

Cevap

The value of

h(11)

is 26.

The correct answer is 26. Since the function

h

is linear, its rate of change (slope) is constant. A slope of 3 means that for every 1-unit increase in

x

, the value of

h(x)

increases by 3. The change in

x

from 5 to 11 is

11 - 5 = 6

. Therefore, the corresponding increase in

h(x)

is

3 \times 6 = 18

. Adding this increase to the known value of

h(5) = 8

gives

h(11) = 8 + 18 = 26

.

Adım Adım Çözüm

1

Determine the change in the input variable

x

from the known point to the target point.

The change in

x

is

11 - 5 = 6

.

This identifies how many units the input increases from

x = 5

to

x = 11

.

2

Multiply the change in

x

by the slope to find the corresponding change in the function value.

The change in the function value is

3 \times 6 = 18

.

The slope represents the constant rate of change (change in

y

divided by change in

x

). Therefore, multiplying the slope by the change in

x

gives the total change in the output.

3

Add the calculated change in the function value to the initial function value.

h(11) = 8 + 18 = 26

.

Adding the total increase in output to the starting output value at

x = 5

yields the value of the function at

x = 11

.

Anahtar Kavram

Linear functions have a constant rate of change, which is represented by the slope. The change in the output value is equal to the slope multiplied by the change in the input value.

Alternatif Yöntem

Alternatively, find the equation of the line using the point-slope form:

y - y_1 = m(x - x_1)

. Substituting the slope

m = 3

and the point

(5, 8)

gives

y - 8 = 3(x - 5)

, which simplifies to

y = 3x - 7

. Thus, the function is defined by

h(x) = 3x - 7

. Evaluating this function at

x = 11

yields

h(11) = 3(11) - 7 = 33 - 7 = 26

.

Tahmini Süre:1m 0s

Soru 42Soru →

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(k, 12)$ and $(8, k)$ , where $k$ is a constant. If the slope of the graph of $f$ is $-\frac{1}{3}$ , what is the value of $k$ ?

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Cevap: 14

Cevap

14

The slope of a line passing through

(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

(k, 12)

and

(8, k)

and the slope

-\frac{1}{3}

into the formula gives

-\frac{1}{3} = \frac{k - 12}{8 - k}

. Cross-multiplying yields

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

, which simplifies to

3k - 36 = -8 + k

. Isolating the variable

k

gives

2k = 28

, so

k = 14

.

Adım Adım Çözüm

1

Apply the slope formula

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

using the points

(k, 12)

and

(8, k)

with slope

-\frac{1}{3}

.

-\frac{1}{3} = \frac{k - 12}{8 - k}

The slope of a linear function is constant and defined by the ratio of the change in

y

-values to the change in

x

-values.

2

Cross-multiply to solve the equation for

k

.

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

To eliminate the fractions and solve the rational equation.

3

Distribute and simplify both sides of the equation.

3k - 36 = -8 + k

Distributing the constants on both sides prepares the equation for isolating the variable

k

.

4

Isolate the variable

k

by subtracting

k

from both sides and adding

36

to both sides.

2k = 28

Grouping like terms on opposite sides of the equation.

5

Divide by

2

to find the value of

k

.

k = 14

To find the final numerical value of the constant

k

.

Anahtar Kavram

Linear function slope formula

Tahmini Süre:1m 30s

Soru 43Soru →

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

$x$	$f(x)$
$1$	$5$
$4$	$17$
$7$	$29$

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

4

\frac{1}{4}

3

5

Cevabı ve açıklamayı göster

Cevap: $4$

Cevap

The slope of the graph of

f

in the

xy

-plane is

4

.

The slope of a linear function can be found using any two points from the table,

(x_1, y_1)

and

(x_2, y_2)

, with the formula

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

. Using the points

(1, 5)

and

(4, 17)

, the slope is

\frac{17 - 5}{4 - 1} = 4

.

Adım Adım Çözüm

1

Select two points from the table to calculate the slope.

Using the points

(1, 5)

and

(4, 17)

, we have

x_1 = 1

,

y_1 = 5

,

x_2 = 4

, and

y_2 = 17

.

Any two points on a line can be used to find its constant slope.

2

Apply the slope formula

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

.

m = \frac{17 - 5}{4 - 1}

The slope formula calculates the ratio of the change in the vertical direction (

y

) to the change in the horizontal direction (

x

).

3

Simplify the expression to find the final slope value.

m = \frac{12}{3} = 4

Subtracting the coordinates and dividing gives the simplified slope.

Anahtar Kavram

Calculating the slope of a linear function from a table of values.

Tahmini Süre:45s

Soru 44Soru →

In the $xy$ -plane, the graphs of two linear functions, $f$ and $g$ , are perpendicular lines that intersect at the point $(12, k)$ , where $k$ is a positive constant. If the $y$ -intercept of the graph of $f$ is $(0, 24)$ and the $y$ -intercept of the graph of $g$ is $(0, -6)$ , what is the value of $k$ ?

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Cevap: 18

Cevap

18

To find the value of

k

, we determine the slopes of the two lines. The line representing

f

passes through

(0, 24)

and

(12, k)

, so its slope is

m_f = \frac{k - 24}{12}

. The line representing

g

passes through

(0, -6)

and

(12, k)

, so its slope is

m_g = \frac{k + 6}{12}

. Since the two lines are perpendicular, the product of their slopes is

-1

. This gives the equation

\left(\frac{k - 24}{12}\right)\left(\frac{k + 6}{12}\right) = -1

. Multiplying both sides by

144

yields

(k - 24)(k + 6) = -144

. Expanding the left side gives

k^2 - 18k - 144 = -144

. Adding

144

to both sides results in

k^2 - 18k = 0

, which factors as

k(k - 18) = 0

. Since

k

is a positive constant,

k

must be

18

.

Adım Adım Çözüm

1

Determine the slopes of the lines

f

and

g

in terms of

k

.

The slope of

f

is

m_f = \frac{k - 24}{12 - 0} = \frac{k - 24}{12}

. The slope of

g

is

m_g = \frac{k - (-6)}{12 - 0} = \frac{k + 6}{12}

.

We use the slope formula

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

for each line passing through their respective

y

-intercepts and their intersection point

(12, k)

.

2

Set up an equation using the perpendicular condition.

m_f \cdot m_g = -1 \implies \left(\frac{k - 24}{12}\right)\left(\frac{k + 6}{12}\right) = -1 \implies \frac{(k - 24)(k + 6)}{144} = -1

Since the graphs of

f

and

g

are perpendicular, the product of their slopes must equal

-1

.

3

Solve the quadratic equation for

k

.

(k - 24)(k + 6) = -144 \implies k^2 - 18k - 144 = -144 \implies k^2 - 18k = 0 \implies k(k - 18) = 0

. Since

k

must be positive,

k = 18

.

Multiplying by

144

, expanding, and factoring the quadratic equation gives the possible values

k = 0

and

k = 18

. The problem states

k

is positive, so

k = 18

.

Anahtar Kavram

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

-1

.

Soru 45Soru →

A linear model is used to estimate the height of a plant, in centimeters, based on the number of weeks since it was planted. According to the model, the plant's height is $5$ centimeters at week $2$ , and its height is $8$ centimeters at week $8$ . Which of the following is the predicted height of the plant, in centimeters, at week $16$ ?

14

29

12

47

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Cevap: $12$

Cevap

The predicted height of the plant at week 16 is 12 centimeters.

The correct answer is 12. To find the predicted height, we determine the linear model equation. The slope is calculated as

m = \frac{8 - 5}{8 - 2} = \frac{3}{6} = 0.5

. Substituting the point

(2, 5)

into the slope-intercept form

y = mx + b

gives

5 = 0.5(2) + b

, which simplifies to

b = 4

. The linear model is

y = 0.5x + 4

. Substituting

16

for

x

yields

y = 0.5(16) + 4 = 8 + 4 = 12

.

Adım Adım Çözüm

1

Identify two data points from the given information to represent the linear relationship.

The coordinates are

(2, 5)

and

(8, 8)

where the x-coordinate represents the week and the y-coordinate represents the height of the plant in centimeters.

A linear relationship is uniquely determined by two points, which allows us to find the slope and equation of the line.

2

Calculate the slope of the line, which represents the constant growth rate of the plant.

The slope

m

is

\frac{8 - 5}{8 - 2} = \frac{3}{6} = 0.5

centimeters per week.

The slope represents the constant rate of change of the plant's height per week.

3

Determine the y-intercept of the linear model using one of the coordinates.

Using the point

(2, 5)

and the equation

y = mx + b

, we get

5 = 0.5(2) + b

, which simplifies to

b = 4

.

Finding the y-intercept allows us to write the complete equation for the linear function representing the plant's growth.

4

Substitute the target week into the linear function to predict the height of the plant.

For week 16,

y = 0.5(16) + 4 = 8 + 4 = 12

centimeters.

Evaluating the function at

x = 16

yields the predicted height of the plant at that specific time.

Anahtar Kavram

Finding and evaluating linear functions using two given coordinate points.

Tahmini Süre:1m 30s

Soru 46Soru →

For a linear function $f$ , the equation $f(x + 3) - f(x - 1) = 16$ is true for all real numbers $x$ . If the graph of $y = f(x)$ in the $xy$ -plane passes through the point $(2, 5)$ , what is the $x$ -intercept of the graph of $f$ ?

-\frac{3}{4}

\frac{3}{4}

\frac{11}{8}

-18

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Cevap: $\frac{3}{4}$

Cevap

The

x

-intercept of the graph of

f

is

\frac{3}{4}

.

By writing the linear function as

f(x) = mx + b

, we can evaluate

f(x+3) - f(x-1)

as

(mx + 3m + b) - (mx - m + b) = 4m

. Setting this equal to the given value of

16

yields a slope of

m = 4

. Using the point

(2, 5)

in

f(2) = 4(2) + b = 5

gives the

y

-intercept

b = -3

, so the function is

f(x) = 4x - 3

. The

x

-intercept is found by setting

f(x) = 0

, which yields

x = \frac{3}{4}

.

Adım Adım Çözüm

1

Represent the linear function in slope-intercept form

f(x) = mx + b

and substitute the expressions

x+3

and

x-1

into the function.

f(x+3) = m(x+3) + b

and

f(x-1) = m(x-1) + b

.

This allows us to evaluate the given functional equation using the general parameters of a linear function.

2

Calculate the difference

f(x+3) - f(x-1)

and set it equal to

16

to find the slope

m

.

(mx + 3m + b) - (mx - m + b) = 4m = 16

, which simplifies to

m = 4

.

The difference between two values of a linear function is directly proportional to the difference in their inputs, where the constant of proportionality is the slope.

3

Substitute the point

(2, 5)

and the slope

m = 4

into

f(x) = 4x + b

to find the

y

-intercept

b

.

5 = 4(2) + b \implies 5 = 8 + b \implies b = -3

.

A linear function is completely defined once both its slope and a point on its graph are known.

4

Find the

x

-intercept of the graph by setting

f(x) = 0

and solving for

x

.

4x - 3 = 0 \implies x = \frac{3}{4}

.

The

x

-intercept of a function's graph is the input value at which the function's output equals zero.

Anahtar Kavram

Determining the slope, equation, and intercepts of a linear function from functional relationships and coordinate points.

Tahmini Süre:2m 0s

Soru 47Soru →

A water tank is being filled at a constant rate. The volume of water in the tank, in gallons, is a linear function of the time, in minutes, since the filling process began. The volume of water in the tank was $24$ gallons after $3$ minutes of filling, and it was $40$ gallons after $7$ minutes of filling. What was the initial volume of water, in gallons, in the tank before the filling process began?

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Cevap: 12

Cevap

The initial volume of water in the tank was 12 gallons.

To find the initial volume of water, we must determine the y-intercept of the linear relationship between the volume and time. First, find the rate of change (slope) using the formula m = \frac{V_2 - V_1}{t_2 - t_1}. Substituting the given values gives m = \frac{40 - 24}{7 - 3} = \frac{16}{4} = 4. Using the slope-intercept form V(t) = mt + b, substitute one of the points, such as (3, 24), to solve for b: 24 = 4(3) + b, which simplifies to b = 12. Thus, the initial volume of water is 12 gallons.

Adım Adım Çözüm

1

Calculate the slope (rate of change) of the volume with respect to time.

4

The slope is the change in volume divided by the change in time: \frac{40 - 24}{7 - 3} = \frac{16}{4} = 4.

2

Write the linear function V(t) = mt + b using the calculated slope, and substitute one of the given points to solve for b.

b = 12

Using the point (3, 24) and the slope m = 4, we substitute into V(t) = 4t + b to get 24 = 4(3) + b, which simplifies to 24 = 12 + b, meaning b = 12.

3

Identify the initial volume of water in the tank, which is the value of V(t) when t = 0.

12

When t = 0, V(0) = 4(0) + 12 = 12, which represents the initial volume.

Anahtar Kavram

Determining the y-intercept (initial value) of a linear function given two points.

Soru 48Soru →

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 positive constants. The line $y = -2x$ is perpendicular to the line that passes through the origin $(0,0)$ and the midpoint of the segment connecting the two intercepts of $f$ . If $r = 12$ , what is the value of $s$ ?

Cevabı ve açıklamayı göster

Cevap: 24

Cevap

24

The midpoint of the segment connecting

(0, 12)

and

(s, 0)

is

(\frac{s}{2}, 6)

. The line passing through the origin

(0, 0)

and this midpoint has a slope of

\frac{6}{s/2} = \frac{12}{s}

. Since this line is perpendicular to the line

y = -2x

, which has a slope of

-2

, its slope must be the negative reciprocal of

-2

, which is

\frac{1}{2}

. Equating these two slopes, we get

\frac{12}{s} = \frac{1}{2}

, which simplifies to

s = 24

.

Adım Adım Çözüm

1

Find the coordinates of the intercepts and their midpoint.

The intercepts are

(0, 12)

and

(s, 0)

, and their midpoint is

(\frac{s}{2}, 6)

.

The intercepts of the function

f

form a line segment whose midpoint must be calculated.

2

Calculate the slope of the line passing through the origin and the midpoint.

The slope is

\frac{12}{s}

.

A line passing through the origin

(0,0)

and a point

(x_1, y_1)

has a slope of

\frac{y_1}{x_1}

.

3

Relate the slope of the line to the perpendicular line

y = -2x

.

The slope of the line must be

\frac{1}{2}

.

Perpendicular lines have slopes that are negative reciprocals of each other, and the negative reciprocal of

-2

is

\frac{1}{2}

.

4

Solve for the value of

s

.

s = 24

.

Setting the two expressions for the slope equal,

\frac{12}{s} = \frac{1}{2}

, yields

s = 24

.

Anahtar Kavram

Linear Functions and Graphs

Tahmini Süre:2m 30s

Soru 49Soru →

In the $xy$ -plane, the graph of the linear function $f$ passes through the point $(2, 5)$ and has a $y$ -intercept of $(0, b)$ , where $b$ is a constant. The function $g$ is defined by $g(x) = f(x) - 4$ . If the $x$ -intercept of the graph of $g$ is $(6, 0)$ , what is the value of $b$ ?

10

\frac{19}{2}

\frac{11}{2}

\frac{11}{4}

Cevabı ve açıklamayı göster

Cevap: $\frac{11}{2}$

Cevap

The correct value of

b

is

\frac{11}{2}

.

The correct answer is the value

\frac{11}{2}

. By writing

f(x) = mx + b

and using the point

(2, 5)

, we get

m = \frac{5-b}{2}

. Applying the definition of

g(x) = f(x) - 4

gives

g(x) = mx + b - 4

. Since the graph of

g

has an

x

-intercept of

(6, 0)

, substituting

x = 6

and

g(6) = 0

gives

6m + b - 4 = 0

. Substituting the expression for

m

yields

3(5-b) + b - 4 = 0

, which simplifies to

11 - 2b = 0

, and thus

b = \frac{11}{2}

.

Adım Adım Çözüm

1

Write the linear function

f(x)

in slope-intercept form using the given

y

-intercept

(0, b)

.

f(x) = mx + b

, where

m

is the slope of the line.

This sets up the general equation of the line with unknown parameters

m

and

b

.

2

Substitute the coordinates of the point

(2, 5)

into the equation for

f(x)

to express

m

in terms of

b

.

5 = 2m + b

, which simplifies to

m = \frac{5 - b}{2}

.

Since the point lies on the graph of

f

, its coordinates must satisfy the equation.

3

Define

g(x)

using the relationship

g(x) = f(x) - 4

and apply the

x

-intercept

(6, 0)

.

g(x) = mx + b - 4

. Since the

x

-intercept is

(6, 0)

, we have

g(6) = 0

, which gives

6m + b - 4 = 0

.

The

x

-intercept is the point where the output of the function is zero.

4

Substitute the expression for

m

from Step 2 into the equation from Step 3 and solve for

b

.

6\left(\frac{5-b}{2}\right) + b - 4 = 0 \Rightarrow 3(5-b) + b - 4 = 0 \Rightarrow 15 - 3b + b - 4 = 0 \Rightarrow 11 - 2b = 0 \Rightarrow b = \frac{11}{2}

.

This solves the system of equations to determine the value of the constant

b

.

Anahtar Kavram

Linear Functions and Graphs

Soru 50Soru →

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

$x$	$h(x)$
$2$	$k - 4$
$5$	$k + 8$
$8$	$2k + 2$

If $k$ is a constant, what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

The value of the constant

k

is

18

.

Since the function

h

is linear, its rate of change (slope) is constant. Calculating the slope using the first two coordinate pairs

(2, k - 4)

and

(5, k + 8)

gives

\frac{(k + 8) - (k - 4)}{5 - 2} = \frac{12}{3} = 4

. Using the next two coordinate pairs

(5, k + 8)

and

(8, 2k + 2)

gives the slope as

\frac{(2k + 2) - (k + 8)}{8 - 5} = \frac{k - 6}{3}

. Setting these two slope values equal to each other gives the equation

\frac{k - 6}{3} = 4

. Multiplying both sides by

3

results in

k - 6 = 12

, and adding

6

to both sides yields

k = 18

.

Adım Adım Çözüm

1

Identify that the rate of change (slope) of a linear function is constant between any two points.

The slope calculated from the first two points must equal the slope calculated from the second and third points.

This relationship allows us to set up an algebraic equation to solve for the unknown constant

k

.

2

Calculate the slope using the first two points:

(2, k - 4)

and

(5, k + 8)

.

Slope =

\frac{(k + 8) - (k - 4)}{5 - 2} = \frac{k + 8 - k + 4}{3} = \frac{12}{3} = 4

.

This simplifies to a constant numerical value of

4

for the slope of the function.

3

Calculate the slope using the second and third points:

(5, k + 8)

and

(8, 2k + 2)

.

Slope =

\frac{(2k + 2) - (k + 8)}{8 - 5} = \frac{2k + 2 - k - 8}{3} = \frac{k - 6}{3}

.

This provides a second expression for the slope in terms of the variable

k

.

4

Equate the two slope expressions and solve for

k

.

\frac{k - 6}{3} = 4 \implies k - 6 = 12 \implies k = 18

.

Setting the two expressions equal and solving isolating

k

gives the correct value of

18

.

Anahtar Kavram

A linear function has a constant rate of change (slope) between any two points on its graph.

Soru 51Soru →

The table below shows some values for a linear function $f$ , where $k$ is a positive constant.

$x$	$f(x)$
$0$	$3$
$k$	$2k + 6$
$4k$	$27$

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

2

-4

4

\frac{1}{4}

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The slope of the graph of

f

is

4

.

The correct answer is

4

. A linear function can be written in the form

f(x) = mx + b

, where

m

is the slope and

b

is the

y

-intercept. Since the

y

-intercept of the graph of

f

is

(0, 3)

, we have

b = 3

, so the function is

f(x) = mx + 3

. Using the given points, we can set up the equations

f(k) = mk + 3 = 2k + 6

and

f(4k) = 4mk + 3 = 27

. Solving this system yields

mk = 6

and

2k = 3

, which gives

k = 1.5

. Substituting

k = 1.5

into

mk = 6

gives the slope

m = 4

.

Adım Adım Çözüm

1

Express the linear function

f(x)

using the slope-intercept form and the given

y

-intercept.

f(x) = mx + 3

, where

m

is the slope and the

y

-intercept is

(0, 3)

.

A linear function has the general form

f(x) = mx + b

. The table indicates that when

x = 0

,

f(x) = 3

, which gives the

y

-intercept

(0, 3)

and establishes

b = 3

.

2

Use the point

(k, 2k + 6)

from the table to write an equation involving

m

and

k

.

mk + 3 = 2k + 6 \implies mk = 2k + 3

.

Substituting

x = k

into the function expression gives

f(k) = mk + 3

. Setting this equal to the table value

2k + 6

allows us to express

mk

in terms of

k

.

3

Use the point

(4k, 27)

from the table to write another equation involving

m

and

k

, and solve for the product

mk

.

4mk + 3 = 27 \implies 4mk = 24 \implies mk = 6

.

Substituting

x = 4k

into the function expression gives

f(4k) = 4mk + 3

. Setting this equal to the table value

27

allows us to solve directly for the numerical value of

mk

.

4

Substitute the value of

mk

into the equation from Step 2 to solve for the constant

k

.

6 = 2k + 3 \implies 2k = 3 \implies k = 1.5

.

By replacing

mk

with

6

in the equation

mk = 2k + 3

, we get a single-variable linear equation that we can solve for

k

.

5

Solve for the slope

m

using the value of

k

and the product

mk

.

m(1.5) = 6 \implies m = 4

.

Since

mk = 6

and

k = 1.5

, dividing the product

6

by

1.5

yields the slope

m

.

Anahtar Kavram

Using coordinate points and intercepts to determine the slope of a linear function represented in a table.

Soru 52Soru →

The linear function $f$ is defined such that its graph in the $xy$ -plane is perpendicular to the line $3x - 4y = 20$ . The graph of another linear function, $g$ , is the result of shifting the graph of $f$ right by $5$ units and down by $4$ units. If $f(0) = 8$ and the graph of $g$ intersects the $x$ -axis at the point $(k, 0)$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

The slope of the line

3x - 4y = 20

is

\frac{3}{4}

. The slope of a line perpendicular to it is the negative reciprocal,

-\frac{4}{3}

. With a

y

-intercept of

(0, 8)

, the function is

f(x) = -\frac{4}{3}x + 8

. Translating this function

5

units right and

4

units down yields

g(x) = f(x - 5) - 4 = -\frac{4}{3}(x - 5) + 4 = -\frac{4}{3}x + \frac{32}{3}

. Setting

g(k) = 0

to find the

x

-intercept gives

-\frac{4}{3}k + \frac{32}{3} = 0

, which solves to

k = 8

.

Adım Adım Çözüm

1

Find the slope of the line

3x - 4y = 20

.

The slope is

\frac{3}{4}

.

Converting the line equation to slope-intercept form

y = mx + b

gives

y = \frac{3}{4}x - 5

, where the coefficient of

x

represents the slope.

2

Find the slope of

f

.

The slope of

f

is

-\frac{4}{3}

.

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

3

Write the function

f(x)

.

f(x) = -\frac{4}{3}x + 8

.

Since the

y

-intercept is given by

f(0) = 8

and the slope is

-\frac{4}{3}

, the slope-intercept form is

f(x) = -\frac{4}{3}x + 8

.

4

Determine the function

g(x)

by translating

f(x)

5

units right and

4

units down.

g(x) = -\frac{4}{3}x + \frac{32}{3}

.

A translation of

f(x)

right by

5

units and down by

4

units corresponds to

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

. Substituting

x-5

into

f(x)

gives

g(x) = -\frac{4}{3}(x - 5) + 8 - 4 = -\frac{4}{3}x + \frac{32}{3}

.

5

Set

g(k) = 0

and solve for

k

.

k = 8

.

The graph of

g

intersects the

x

-axis at

(k, 0)

, which means

g(k) = 0

. Solving

-\frac{4}{3}k + \frac{32}{3} = 0

yields

k = 8

.

Anahtar Kavram

Writing linear functions using perpendicular slopes and performing vertical and horizontal transformations.

Soru 53Soru →

In the $xy$ -plane, the graph of a linear function $f$ passes through the point $(1, 3)$ and has a slope of $m$ , where $m \neq 0$ . The graph of a second linear function, $g$ , is obtained by reflecting the graph of $f$ across the $y$ -axis and then translating it up by $4$ units. If the graphs of $f$ and $g$ intersect at a point on the $x$ -axis, what is the value of $m$ ?

1

-2

5

3

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The correct value is

5

. By using the point-slope form, the first function is

f(x) = mx - m + 3

. Reflecting it across the

y

-axis gives

f(-x) = -mx - m + 3

, and translating it up by

4

units yields

g(x) = -mx - m + 7

. Since they intersect on the

x

-axis, the

y

-value at their intersection point is

0

. This gives the system of equations

mx_0 = m - 3

and

-mx_0 = m - 7

. Adding these equations eliminates the

mx_0

term, leaving

2m - 10 = 0

, which simplifies to

m = 5

.

Adım Adım Çözüm

1

Write the equation of the linear function

f(x)

using the point-slope form.

f(x) = m(x - 1) + 3 = mx - m + 3

The function passes through the point

(1, 3)

and has a slope of

m

.

2

Determine the equation of the linear function

g(x)

by applying the reflection and translation transformations.

g(x) = f(-x) + 4 = -mx - m + 7

Reflecting across the

y

-axis replaces

x

with

-x

, and translating up by

4

units adds

4

to the function.

3

Set the

y

-coordinates of the intersection point

(x_0, 0)

on the

x

-axis to

0

for both functions.

mx_0 - m + 3 = 0 \implies mx_0 = m - 3

and

-mx_0 - m + 7 = 0 \implies -mx_0 = m - 7

An intersection on the

x

-axis means that the

y

-coordinate of the intersection point is

0

.

4

Solve the system of equations to find the value of

m

.

0 = (m - 3) + (m - 7) \implies 2m = 10 \implies m = 5

Adding the two equations eliminates the term

mx_0

, allowing us to solve directly for

m

.

Anahtar Kavram

Applying transformations (reflections and translations) to linear functions and finding their intercepts in the coordinate plane.

Soru 54Soru →

For the linear function $f$ , the graph of $y = f(x)$ has a slope of $3$ and contains the point $(2, 7)$ . The graph of another linear function, $g$ , is parallel to the graph of $f$ . If $g(0) = -4$ , what is the value of $g(5)$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

11

Since the graph of

g

is parallel to the graph of

f

, it has the same slope of

3

. The value

g(0) = -4

indicates that the

y

-intercept of the graph of

g

is

-4

. Thus, the equation for

g(x)

is

g(x) = 3x - 4

. Evaluating this function at

x = 5

gives

g(5) = 3(5) - 4 = 11

.

Adım Adım Çözüm

1

Determine the slope of the function

g

.

The slope of

g

is

3

.

Parallel lines in the coordinate plane have equal slopes, and the slope of the graph of

f

is given as

3

.

2

Write the equation for

g(x)

.

g(x) = 3x - 4

Using the slope-intercept form

g(x) = mx + b

with slope

m = 3

and

y

-intercept

b = g(0) = -4

.

3

Evaluate

g(5)

.

11

Substitute

x = 5

into the equation for

g(x)

to get

g(5) = 3(5) - 4 = 11

.

Anahtar Kavram

Parallel lines have the same slope. A linear function can be written in slope-intercept form,

y = mx + b

, where

m

is the slope and

b

is the

y

-intercept.

Soru 55Soru →

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(-2, 5)$ and $(4, 1)$ . If the linear function $g$ is defined by $g(x) = f(x) + 3$ , what is the $x$ -intercept of the graph of $g$ ?

6

\frac{11}{2}

10

\frac{20}{3}

Cevabı ve açıklamayı göster

Cevap: $10$

Cevap

10

The correct answer is

10

. First, find the slope of

f

using the formula

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

, which gives

m = \frac{1 - 5}{4 - (-2)} = -\frac{2}{3}

. Using the point

(4, 1)

in the point-slope formula, the equation of

f

is

f(x) = -\frac{2}{3}x + \frac{11}{3}

. Since

g(x) = f(x) + 3

, the equation of

g

is

g(x) = -\frac{2}{3}x + \frac{20}{3}

. Setting

g(x) = 0

to find the

x

-intercept gives

x = 10

.

Adım Adım Çözüm

1

Calculate the slope of the linear function

f

m = -\frac{2}{3}

The slope

m

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, 1)

yields

m = \frac{1 - 5}{4 - (-2)} = \frac{-4}{6} = -\frac{2}{3}

.

2

Find the equation of function

f

f(x) = -\frac{2}{3}x + \frac{11}{3}

Using the point-slope form

y - y_1 = m(x - x_1)

with the point

(4, 1)

, we get

y - 1 = -\frac{2}{3}(x - 4) \Rightarrow y = -\frac{2}{3}x + \frac{8}{3} + 1 = -\frac{2}{3}x + \frac{11}{3}

.

3

Determine the equation of function

g

g(x) = -\frac{2}{3}x + \frac{20}{3}

Since

g(x) = f(x) + 3

, we add

3

to the expression for

f(x)

:

g(x) = -\frac{2}{3}x + \frac{11}{3} + 3 = -\frac{2}{3}x + \frac{20}{3}

.

4

Find the

x

-intercept of the graph of

g

x = 10

The

x

-intercept is the value of

x

when

g(x) = 0

. Setting

g(x) = 0

gives

0 = -\frac{2}{3}x + \frac{20}{3} \Rightarrow \frac{2}{3}x = \frac{20}{3} \Rightarrow x = 10

.

Anahtar Kavram

Finding the equation of a linear function from two points and applying vertical translations to find key features such as intercepts.

Tahmini Süre:1m 30s

Soru 56Soru →

A state department of transportation models the relationship between the age of a highway, in years, and its road quality index on a scale from $0$ to $100$ . The index decreases at a constant rate with respect to the age of the highway. At age $4$ years, the highway has a road quality index of $82$ . At age $12$ years, the highway has a road quality index of $66$ . According to the model, after how many years from its construction will the highway's road quality index reach $50$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

According to the model, the highway's road quality index will reach

50

after

20

years.

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

(4, 82)

and

(12, 66)

, the slope is

\frac{66 - 82}{12 - 4} = -2

. The linear function can be written in point-slope form as

I - 82 = -2(t - 4)

, which simplifies to

I = -2t + 90

. Setting

I = 50

yields the equation

50 = -2t + 90

. Subtracting

90

from both sides gives

-40 = -2t

, and dividing by

-2

results in

t = 20

. Therefore, the index reaches

50

after

20

years.

Adım Adım Çözüm

1

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

The slope is

-2

.

Since the index decreases at a constant rate, the relationship is linear. The slope is calculated as the change in the road quality index divided by the change in the highway's age:

\frac{66 - 82}{12 - 4} = \frac{-16}{8} = -2

.

2

Determine the linear equation relating the road quality index,

I

, and the age,

t

.

I = -2t + 90

Using the point-slope formula with the point

(4, 82)

and slope

-2

:

I - 82 = -2(t - 4) \Rightarrow I - 82 = -2t + 8 \Rightarrow I = -2t + 90

.

3

Solve for the age,

t

, when the index

I

is

50

.

t = 20

Substitute

50

for

I

in the equation:

50 = -2t + 90

. Subtract

90

from both sides to get

-40 = -2t

. Divide by

-2

to find

t = 20

.

Anahtar Kavram

Writing and solving linear equations from two points

Soru 57Soru →

A hot air balloon is at an altitude of $h$ meters. The balloon begins to descend at a constant rate. After $3$ minutes, the altitude of the balloon is $540$ meters. After $8$ minutes, the altitude of the balloon is $390$ meters. If the altitude of the balloon is modeled by a linear function of time, what was the initial altitude of the balloon, in meters?

Cevabı ve açıklamayı göster

Cevap: 630

Cevap

The initial altitude of the balloon was 630 meters.

To find the initial altitude, we model the balloon's descent as a linear equation of the form

h(t) = mt + b

, where

h(t)

is the altitude at time

t

,

m

is the constant rate of change, and

b

is the initial altitude. Using the given points

(3, 540)

and

(8, 390)

, we find the slope

m = \frac{390 - 540}{8 - 3} = -30

meters per minute. Substituting

m = -30

and the point

(3, 540)

into

h(t) = mt + b

gives

540 = -30(3) + b

. Solving for

b

yields

b = 540 + 90 = 630

meters.

Adım Adım Çözüm

1

Identify two data points from the problem context

The coordinate points are

(3, 540)

and

(8, 390)

To define the linear relationship, we need at least two coordinate points representing (time, altitude)

2

Calculate the slope (

m

) of the linear function

m = \frac{390 - 540}{8 - 3} = \frac{-150}{5} = -30

The rate of change represents the speed at which the balloon descends each minute

3

Solve for the vertical intercept (

b

) using the slope-intercept form

y = mx + b

540 = -30(3) + b \implies b = 630

The initial altitude corresponds to the altitude at time

t = 0

, which is the vertical intercept of the linear function

Anahtar Kavram

Linear Functions and Graphs

Soru 58Soru →

An environmental scientist is monitoring the water level of a reservoir during a dry season. The water level, $L(d)$ , in meters, can be modeled by a linear function of the number of days, $d$ , since the start of the dry season. On day 12, the water level was 30 meters, and on day 20, the water level was 26 meters. If the water level continues to decrease at this constant rate, on which day will the water level be exactly 18 meters?

Day 18

Day 24

Day 36

Day 40

Cevabı ve açıklamayı göster

Cevap: Day 36

Cevap

Day 36

To find the day when the water level is exactly 18 meters, we first find the constant rate of change (slope) using the given points

(12, 30)

and

(20, 26)

. The slope is

m = \frac{26 - 30}{20 - 12} = -0.5

meters per day. Using the point-slope form with

(12, 30)

, we get

L(d) - 30 = -0.5(d - 12)

, which simplifies to

L(d) = -0.5d + 36

. Substituting 18 for

L(d)

gives

18 = -0.5d + 36

. Subtracting 36 from both sides results in

-18 = -0.5d

, and dividing by

-0.5

yields

d = 36

. Thus, the correct answer is Day 36.

Adım Adım Çözüm

1

Calculate the constant rate of change (slope,

m

) using the points

(12, 30)

and

(20, 26)

.

m = \frac{26 - 30}{20 - 12} = \frac{-4}{8} = -0.5

meters per day.

A linear function has a constant rate of change, which is determined by the slope formula:

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

.

2

Write the equation of the linear model in point-slope form and solve for the standard form.

L(d) - 30 = -0.5(d - 12) \Rightarrow L(d) = -0.5d + 36

.

Using the point-slope formula

y - y_1 = m(x - x_1)

with the point

(12, 30)

and slope

-0.5

defines the function for any day

d

.

3

Substitute

L(d) = 18

into the equation and solve for

d

.

18 = -0.5d + 36 \Rightarrow -18 = -0.5d \Rightarrow d = 36

.

Setting the dependent variable to the target water level of 18 meters yields the corresponding day.

Anahtar Kavram

Determining and evaluating linear equations from two given coordinate points

Alternatif Yöntem

Instead of constructing the entire equation, you can analyze the constant rate of change directly. The water level decreases by

4

meters (from 30 to 26) over

8

days (from day 12 to day 20), which means the rate of decrease is

0.5

meters per day. To drop from 26 meters (on day 20) to 18 meters requires a further decrease of

26 - 18 = 8

meters. At a rate of

0.5

meters per day, dropping 8 meters will take

\frac{8}{0.5} = 16

days. Adding these 16 days to day 20 yields day 36.

Tahmini Süre:1m 30s

Soru 59Soru →

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

$x$	$f(x)$
$-2$	$13$
$2$	$5$
$6$	$-3$

If the graph of a second linear function, $g$ , has a slope that is twice the slope of the graph of $f$ and passes through the point $(1, 4)$ , what is the value of $g(5)$ ?

0

-20

-12

-16

Cevabı ve açıklamayı göster

Cevap: -12

Cevap

-12

The correct answer is

-12

. First, calculate the slope of the linear function

f

using two coordinate pairs from the table, such as

(-2, 13)

and

(2, 5)

. The slope is

\frac{5 - 13}{2 - (-2)} = -2

. Since the slope of function

g

is twice the slope of

f

, the slope of

g

is

2 \times (-2) = -4

. Using the point-slope equation with the point

(1, 4)

, we have

g(x) - 4 = -4(x - 1)

, which simplifies to

g(x) = -4x + 8

. Finally, evaluating the function at

x = 5

gives

g(5) = -4(5) + 8 = -12

.

Adım Adım Çözüm

1

Calculate the slope of the linear function

f

using two points from the table,

(-2, 13)

and

(2, 5)

.

The slope of

f

is

m_f = \frac{5 - 13}{2 - (-2)} = \frac{-8}{4} = -2

.

Finding the slope of the reference function is necessary to determine the slope of the second function.

2

Determine the slope of the function

g

.

The slope of

g

is

m_g = 2 \times (-2) = -4

.

The problem states that the slope of

g

is twice the slope of

f

.

3

Find the equation of

g(x)

using the point-slope form with the point

(1, 4)

.

g(x) - 4 = -4(x - 1) \implies g(x) = -4x + 8

.

Using a point on the line and its slope allows us to construct the full function definition.

4

Evaluate

g(5)

by substituting

x = 5

into the equation for

g(x)

.

g(5) = -4(5) + 8 = -12

.

This yields the final value requested by the question.

Anahtar Kavram

Linear Functions and Graphs

Soru 60Soru →

A local coffee shop tracks its remaining syrup inventory at the end of each day. On day 3, the shop has 20 liters of syrup remaining. On day 7, the shop has 12 liters of syrup remaining. If the amount of syrup decreases at a constant daily rate, on which day will the shop have exactly 6 liters of syrup remaining?

4

7

10

31

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10

To find the day when

6

liters of syrup remain, we first find the constant rate of decrease (slope) of the syrup. The slope is calculated as the change in syrup volume divided by the change in days:

m = \frac{12 - 20}{7 - 3} = -2

liters per day. Using the point-slope formula with the point

(3, 20)

, the relationship is

y - 20 = -2(x - 3)

, which simplifies to

y = -2x + 26

. Setting the remaining syrup

y

to

6

gives

6 = -2x + 26

. Solving for

x

yields

2x = 20

, so

x = 10

. Thus, the shop will have exactly

6

liters of syrup remaining on day

10

.

Adım Adım Çözüm

1

Calculate the constant daily rate of decrease (slope) of the syrup inventory.

The rate of decrease is

-2

liters per day.

The slope

m

of a linear relationship between day

x

and remaining syrup

y

is given by

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 20}{7 - 3} = \frac{-8}{4} = -2

.

2

Determine the linear equation modeling the remaining syrup as a function of the day.

The linear equation is

y = -2x + 26

.

Using the point-slope form with the point

(3, 20)

and slope

m = -2

, we have

y - 20 = -2(x - 3)

. Distributing and simplifying gives

y - 20 = -2x + 6

, which simplifies to

y = -2x + 26

.

3

Find the day when the remaining syrup is exactly

6

liters.

The shop will have exactly

6

liters of syrup remaining on day

10

.

Substitute

y = 6

into the equation and solve for

x

:

6 = -2x + 26 \Rightarrow -20 = -2x \Rightarrow x = 10

.

Anahtar Kavram

Linear Functions and Graphs

Alternatif Yöntem

Instead of finding the full linear equation, we can use the rate of change directly. The syrup decreases by

2

liters per day. From day 7, where there are

12

liters remaining, we need to reach

6

liters remaining. This is a decrease of

12 - 6 = 6

liters. Since the rate of decrease is

2

liters per day, it will take

\frac{6}{2} = 3

additional days. Adding

3

days to day 7 gives

7 + 3 = 10

.

Tahmini Süre:1m 30s

Linear Functions and Graphs

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Sık Yapılan Hatalar

Alternatif Yöntem

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Alternatif Yöntem

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Sık Yapılan Hatalar

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