Linear Functions and Graphs Alıştırma Soruları — SAT — Sayfa 2

Soru 21Soru →

In the $xy$ -plane, line $l$ passes through the point $(3, 4)$ and has a slope of $m$ , where $m > 0$ . Line $k$ is perpendicular to line $l$ and passes through the point $(2, 2)$ . If the sum of the $y$ -intercepts of line $l$ and line $k$ is $1$ , what is the value of $m$ ?

\frac{1}{3}

\frac{2}{3}

1

2

Cevabı ve açıklamayı göster

Cevap: $2$

Cevap

The correct value of

m

is

2

.

The correct answer is the value

2

. By finding the equations of both lines in slope-intercept form, we express their

y

-intercepts in terms of

m

: the

y

-intercept of line

l

is

4 - 3m

and the

y

-intercept of line

k

is

\frac{2}{m} + 2

. Setting their sum to

1

gives the equation

3m^2 - 5m - 2 = 0

. Factoring this quadratic yields the solutions

m = -\frac{1}{3}

and

m = 2

. Since

m

must be positive,

m = 2

is the only valid solution.

Adım Adım Çözüm

1

Find the equation and

y

-intercept of line

l

.

Using the point-slope form

y - y_1 = m(x - x_1)

with the point

(3, 4)

and slope

m

, the equation of line

l

is

y - 4 = m(x - 3)

, which simplifies to

y = mx + 4 - 3m

. Thus, the

y

-intercept of line

l

is

4 - 3m

.

Expressing the

y

-intercept of line

l

in terms of

m

allows us to use it in the sum equation.

2

Find the equation and

y

-intercept of line

k

.

Since line

k

is perpendicular to line

l

, its slope is the negative reciprocal of

m

, which is

-\frac{1}{m}

. Using the point-slope form with the point

(2, 2)

, the equation of line

k

is

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

, which simplifies to

y = -\frac{1}{m}x + \frac{2}{m} + 2

. Thus, the

y

-intercept of line

k

is

\frac{2}{m} + 2

.

Expressing the

y

-intercept of line

k

in terms of

m

allows us to use it in the sum equation.

3

Set up the equation for the sum of the

y

-intercepts and solve for

m

.

The sum of the

y

-intercepts is

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

. Simplifying this equation gives

6 - 3m + \frac{2}{m} = 1 \implies 5 - 3m + \frac{2}{m} = 0

. Multiplying by

m

yields

3m^2 - 5m - 2 = 0

. Factoring the quadratic gives

(3m + 1)(m - 2) = 0

, which has solutions

m = -\frac{1}{3}

and

m = 2

. Since

m > 0

, we have

m = 2

.

To determine the unique positive slope that satisfies the given conditions.

Anahtar Kavram

Writing equations of perpendicular lines and finding their intercepts using slopes and points.

Soru 22Soru →

During a chemistry experiment, the temperature of a liquid sample is decreased at a constant rate. At the start of the experiment, the temperature of the liquid is $80^\circ\text{C}$ . After $4$ minutes, the temperature of the liquid is $68^\circ\text{C}$ . If $T(t)$ represents the temperature of the liquid, in degrees Celsius, $t$ minutes after the experiment starts, which of the following equations defines $T$ ?

T(t) = -3t + 80

T(t) = 3t + 80

T(t) = -\frac{1}{3}t + 80

T(t) = 80t - 3

Cevabı ve açıklamayı göster

Cevap: $T(t) = -3t + 80$

Cevap

The equation

T(t) = -3t + 80

correctly defines the temperature function.

The correct equation shows a starting value of

80

and a rate of change of

-3

, which corresponds to the initial temperature of

80^\circ\text{C}

and a decrease of

3^\circ\text{C}

per minute.

Adım Adım Çözüm

1

Identify the

y

-intercept (initial value) of the linear function.

The initial temperature at

t = 0

is

80^\circ\text{C}

, so the

y

-intercept

b

is

80

.

The initial value of a linear relationship in context corresponds to the

y

-intercept.

2

Calculate the slope (constant rate of change) using the coordinate points

(0, 80)

and

(4, 68)

.

The slope

m

is

\frac{68 - 80}{4 - 0} = \frac{-12}{4} = -3

.

The slope is defined as the change in the dependent variable (temperature) divided by the change in the independent variable (time).

3

Substitute the slope

m

and

y

-intercept

b

into the slope-intercept form

T(t) = mt + b

.

T(t) = -3t + 80

Substituting the specific slope and intercept values into the general form defines the particular function.

Anahtar Kavram

Determining a linear equation from a real-world scenario by identifying the initial value and constant rate of change.

Soru 23Soru →

In the $xy$ -plane, line $l$ passes through the points $(-2, -3)$ and $(2, 5)$ . Line $k$ is parallel to line $l$ and has a $y$ -intercept of $(0, -1)$ . If the point $(a, 9)$ lies on line $k$ , what is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

a

is

5

.

To find the value of

a

, we first determine the slope of line

l

using the two given points,

(-2, -3)

and

(2, 5)

. The slope

m

is given by

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

. Since line

k

is parallel to line

l

, it has the same slope of

2

. The

y

-intercept of line

k

is

(0, -1)

, so the equation of line

k

is

y = 2x - 1

. To find the value of

a

, we substitute the point

(a, 9)

into this equation:

9 = 2a - 1

. Solving for

a

gives

10 = 2a

, which simplifies to

a = 5

.

Adım Adım Çözüm

1

Calculate the slope of line

l

using the two given points.

The slope of line

l

is

2

.

Using the slope formula

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

with the points

(-2, -3)

and

(2, 5)

, we get

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

.

2

Find the equation of line

k

.

The equation of line

k

is

y = 2x - 1

.

Line

k

is parallel to line

l

, so it has the same slope,

m = 2

. Its

y

-intercept is

(0, -1)

, which gives the equation

y = 2x - 1

in slope-intercept form.

3

Solve for

a

by substituting the point

(a, 9)

into the equation of line

k

.

The value of

a

is

5

.

Substituting

x = a

and

y = 9

into

y = 2x - 1

gives

9 = 2a - 1

. Solving for

a

yields

10 = 2a

, or

a = 5

.

Anahtar Kavram

Linear Functions and Graphs

Soru 24Soru →

In the $xy$ -plane, line $l_1$ passes through the origin and has a positive slope $m$ , where $m > 1$ . Line $l_2$ is perpendicular to line $l_1$ and has a $y$ -intercept of $(0, 10)$ . The two lines intersect at point $P$ . If the distance from point $P$ to the $y$ -axis is $4$ , what is the value of $m$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2

The equation of line

l_1

is

y = mx

and the equation of line

l_2

is

y = -\frac{1}{m}x + 10

. Setting these equal gives the

x

-coordinate of their intersection as

x = \frac{10m}{m^2 + 1}

. Since the distance from the intersection point to the

y

-axis is

4

, we have

\frac{10m}{m^2 + 1} = 4

. Solving this quadratic equation yields

m = 2

or

m = \frac{1}{2}

. Given that

m > 1

, the correct value is

2

.

Adım Adım Çözüm

1

Write the equations of lines

l_1

and

l_2

.

Line

l_1

has a slope of

m

and passes through

(0,0)

, so its equation is

y = mx

. Line

l_2

is perpendicular to

l_1

, so its slope is

-\frac{1}{m}

. Since its

y

-intercept is

(0,10)

, its equation is

y = -\frac{1}{m}x + 10

.

Setting up the equations of the lines allows us to find their point of intersection.

2

Find the

x

-coordinate of the intersection point

P

.

Equating the two expressions for

y

gives

mx = -\frac{1}{m}x + 10

. Multiplying both sides by

m

yields

m^2 x = -x + 10m

, which simplifies to

(m^2 + 1)x = 10m

, or

x = \frac{10m}{m^2 + 1}

.

The intersection point

P

must satisfy both equations simultaneously.

3

Solve for

m

using the distance from

P

to the

y

-axis.

The distance from

P(x,y)

to the

y

-axis is given by

|x|

. Since

m > 1

,

x

is positive, so the distance is

\frac{10m}{m^2 + 1} = 4

. This simplifies to

10m = 4m^2 + 4

, or

4m^2 - 10m + 4 = 0

. Dividing by

2

gives

2m^2 - 5m + 2 = 0

. Factoring the quadratic yields

(2m - 1)(m - 2) = 0

, giving solutions of

m = \frac{1}{2}

and

m = 2

. Since

m > 1

, the slope of

l_1

must be

2

.

Applying the given distance constraint and slope condition determines the unique value of

m

.

Anahtar Kavram

The relationship between the equations of perpendicular lines and their point of intersection in the coordinate plane.

Soru 25Soru →

The graph of the linear function $f$ in the $xy$ -plane has a $y$ -intercept of $(0, b)$ and an $x$ -intercept of $(a, 0)$ , where $a$ and $b$ are nonzero constants. If $3a = -4b$ , which of the following is the slope of the graph of $f$ ?

-\frac{4}{3}

-\frac{3}{4}

\frac{3}{4}

\frac{4}{3}

Cevabı ve açıklamayı göster

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

Cevap

\frac{3}{4}

The slope of the line passing through

(0, b)

and

(a, 0)

is

m = \frac{0 - b}{a - 0} = -\frac{b}{a}

. Starting with the given equation

3a = -4b

, dividing both sides by

-4a

isolates the slope expression:

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

. Therefore, the slope of the graph of

f

is

\frac{3}{4}

.

Adım Adım Çözüm

1

Identify the coordinates of the intercepts and write the formula for the slope of a line.

The

y

-intercept is

(0, b)

and the

x

-intercept is

(a, 0)

. The slope

m

is given by

m = \frac{0 - b}{a - 0} = -\frac{b}{a}

.

To find the slope, we express it in terms of the variables

a

and

b

using the standard slope formula.

2

Use the given equation to find the ratio

-\frac{b}{a}

.

Divide both sides of the equation

3a = -4b

by

a

to get

3 = -4\left(\frac{b}{a}\right)

. Then, divide both sides by

-4

to get

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

, which means

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

.

We isolate the expression for the slope, which is

-\frac{b}{a}

, using algebraic operations on the given equation.

3

Equate the slope expression to the calculated value.

Since

m = -\frac{b}{a}

and

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

, the slope of the line is

\frac{3}{4}

.

This yields the final value of the slope.

Anahtar Kavram

Calculating the slope of a linear function using intercepts and algebraic substitution.

Soru 26Soru →

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

$x$	$f(x)$
$2$	$7$
$5$	$13$
$8$	$19$

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

2

0.5

3

4

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The slope of the graph of

y = f(x)

is 2.

The slope of a linear function represents the constant rate of change of the function values with respect to the input values. Using any two points from the table, such as

(2, 7)

and

(5, 13)

, the slope

m

can be calculated using the slope formula:

m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{13 - 7}{5 - 2} = \frac{6}{3} = 2

. Therefore, the slope of the graph of

y = f(x)

is

2

.

Adım Adım Çözüm

1

Select two points from the given table.

Two points are

(2, 7)

and

(5, 13)

.

To find the slope of a linear function, we need to calculate the rate of change between any two points on the line.

2

Apply the slope formula

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

.

m = \frac{13 - 7}{5 - 2} = \frac{6}{3} = 2

Calculating the ratio of the vertical change (change in

f(x)

) to the horizontal change (change in

x

) yields the constant slope of the linear function.

Anahtar Kavram

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

Tahmini Süre:1m 0s

Soru 27Soru →

In the $xy$ -plane, the graph of the linear function $f$ is perpendicular to the line with equation $y = 2x - 7$ . If the region in the first quadrant bounded by the graph of $f$ , the $x$ -axis, and the $y$ -axis has an area of $36$ , what is the $x$ -intercept of the graph of $f$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The

x

-intercept of the graph of

f

is

12

.

The correct answer is

12

. Since the graph of

f

is perpendicular to the line

y = 2x - 7

, its slope is

-\frac{1}{2}

. The equation of the line is

f(x) = -\frac{1}{2}x + b

, which has a

y

-intercept of

(0, b)

and an

x

-intercept of

(2b, 0)

. In the first quadrant, these intercepts form a right triangle with the axes, having legs of length

b

and

2b

. The area of this triangle is

\frac{1}{2}(2b)(b) = b^2

. Setting the area equal to

36

gives

b^2 = 36

, so

b = 6

(since

b > 0

). The

x

-intercept is

2b = 2(6) = 12

.

Adım Adım Çözüm

1

Find the slope of the perpendicular line

f

.

The slope of

f

is

-\frac{1}{2}

.

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other. The given line has a slope of

2

, so the slope of

f

must be

-\frac{1}{2}

.

2

Express the

y

-intercept and

x

-intercept of

f

in terms of a single variable.

The

y

-intercept is

(0, b)

and the

x

-intercept is

(2b, 0)

, where

b > 0

.

The equation of

f

can be written as

f(x) = -\frac{1}{2}x + b

. Setting

x = 0

gives the

y

-intercept

b

. Setting

f(x) = 0

and solving for

x

gives the

x

-intercept

2b

.

3

Set up the area equation for the triangle in the first quadrant.

The area is represented by

b^2 = 36

.

The region bounded by the graph of

f

and the coordinate axes in the first quadrant forms a right triangle with perpendicular sides of lengths

b

and

2b

. The area of this triangle is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2b \times b = b^2

.

4

Solve for

b

and calculate the

x

-intercept.

The

x

-intercept is

12

.

Solving

b^2 = 36

with the condition

b > 0

yields

b = 6

. The

x

-intercept is

2b

, which equals

2(6) = 12

.

Anahtar Kavram

Using perpendicular slopes and intercepts of linear functions to analyze geometric areas in the coordinate plane.

Tahmini Süre:2m 0s

Soru 28Soru →

A local municipal water utility charges a flat monthly service fee plus a constant rate for each hundred cubic feet (HCF) of water consumed. During one month, a household that consumed 12 HCF of water was charged 46.00 dollars. Another household that consumed 18 HCF of water was charged 61.00 dollars. What is the flat monthly service fee, in dollars, charged by the utility?

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

The flat monthly service fee is 16 dollars.

By modeling the relationship as a linear function

y = mx + b

, where

x

is the consumption in HCF and

y

is the total charge in dollars, we use the points

(12, 46)

and

(18, 61)

to find the slope

m = \frac{61 - 46}{18 - 12} = 2.5

. Substituting

m = 2.5

and the point

(12, 46)

back into the linear equation gives

46 = 2.5(12) + b

, which simplifies to

46 = 30 + b

. Solving for

b

gives 16, representing the flat service fee.

Adım Adım Çözüm

1

Set up a system of points representing the linear relationship between water consumed and monthly charge.

Two coordinate points are established:

(12, 46)

and

(18, 61)

, where the first coordinate is the consumption in HCF and the second is the charge in dollars.

Since the utility charges a flat fee plus a constant rate, the relationship is linear and can be solved using coordinate points.

2

Find the constant rate per HCF (the slope of the line).

The slope

m

is calculated as

\frac{61 - 46}{18 - 12} = \frac{15}{6} = 2.5

dollars per HCF.

The slope represents the constant rate of change in total cost per HCF of water consumed.

3

Determine the flat monthly service fee (the y-intercept of the line).

Using the slope-intercept equation

y = mx + b

with point

(12, 46)

and

m = 2.5

yields

46 = 2.5(12) + b \implies 46 = 30 + b \implies b = 16

.

The y-intercept represents the cost when consumption is zero, which is the flat monthly service fee.

Anahtar Kavram

Finding the y-intercept of a linear function from two points.

Tahmini Süre:1m 30s

Soru 29Soru →

A scientist is measuring the temperature of a sample that is being heated at a constant rate. At a starting time of $0$ minutes, the temperature of the sample is $12^\circ\text{C}$ . After $8$ minutes of heating, the temperature of the sample is $44^\circ\text{C}$ . If the temperature of the sample increases linearly with time, what is the temperature of the sample, in degrees Celsius, after $15$ minutes of heating?

Cevabı ve açıklamayı göster

Cevap: 72

Cevap

72

The rate of temperature change is

\frac{44 - 12}{8 - 0} = 4^\circ\text{C}

per minute. Using the slope-intercept form, the temperature

T

after

t

minutes can be represented by the function

T(t) = 4t + 12

. Substituting

15

for

t

yields

T(15) = 4(15) + 12 = 60 + 12 = 72

.

Adım Adım Çözüm

1

Calculate the constant rate of temperature change (slope).

4 degrees Celsius per minute

To find how much the temperature increases each minute, divide the total change in temperature by the total change in time:

\frac{44 - 12}{8 - 0} = \frac{32}{8} = 4

.

2

Set up the linear function for temperature

T(t)

over time

t

.

T(t) = 4t + 12

The initial temperature at

t = 0

is

12^\circ\text{C}

, which represents the vertical intercept. The constant rate of change is

4^\circ\text{C}

per minute.

3

Find the temperature at

t = 15

minutes.

72

Substitute

15

for

t

in the equation:

T(15) = 4(15) + 12 = 60 + 12 = 72

.

Anahtar Kavram

Linear Functions and Rates of Change

Tahmini Süre:1m 0s

Soru 30Soru →

A line in the $xy$ -plane has a slope of $-3$ . The line passes through the point $(2, k)$ and has an $x$ -intercept of $(r, 0)$ , where $k$ and $r$ are constants. If $r + k = 10$ , what is the $y$ -intercept of the line?

(0, 4)

(0, 6)

(0, 10)

(0, 12)

Cevabı ve açıklamayı göster

Cevap: $(0, 12)$

Cevap

The y-intercept of the line is

(0, 12)

.

The correct answer is

(0, 12)

. By using the formula for slope

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

with the points

(2, k)

and

(r, 0)

, we obtain

-3 = \frac{k - 0}{2 - r}

. Multiplying both sides by

2 - r

gives

k = -3(2 - r)

, which simplifies to

k = -6 + 3r

, or

3r - k = 6

. We can solve this alongside the given equation

r + k = 10

by adding the two equations:

(3r - k) + (r + k) = 6 + 10 \implies 4r = 16

, which yields

r = 4

. Substituting

r = 4

back into the sum equation gives

k = 6

. Now, using the x-intercept point

(4, 0)

and the slope of

-3

, the equation of the line in point-slope form is

y - 0 = -3(x - 4)

, which simplifies to

y = -3x + 12

. The y-intercept of this line is found by setting

x = 0

, yielding

(0, 12)

.

Adım Adım Çözüm

1

Express the slope of the line using the coordinates of the two given points,

(2, k)

and the x-intercept

(r, 0)

.

Using the slope formula

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

, we set up the equation:

-3 = \frac{k - 0}{2 - r}

.

This connects the geometric concept of slope to the algebraic coordinates of the points on the line.

2

Simplify the slope equation to relate

k

and

r

.

Multiplying both sides by

2 - r

gives

k = -3(2 - r) \implies k = -6 + 3r

, which can be rewritten as

3r - k = 6

.

Simplifying the equation makes it easier to solve as part of a system of linear equations.

3

Solve the system of equations consisting of

3r - k = 6

and the given equation

r + k = 10

.

Adding the two equations together:

(3r - k) + (r + k) = 6 + 10 \implies 4r = 16 \implies r = 4

. Substituting

r = 4

into

r + k = 10

gives

4 + k = 10 \implies k = 6

.

Solving the system of equations determines the numerical values of the constants

r

and

k

.

4

Find the equation of the line using the slope

m = -3

and the x-intercept

(4, 0)

, then determine the y-intercept.

Using point-slope form:

y - 0 = -3(x - 4) \implies y = -3x + 12

. The y-intercept occurs when

x = 0

, giving

y = 12

, which corresponds to the point

(0, 12)

.

Writing the full linear equation allows us to find the y-intercept by evaluating the function at

x = 0

.

Anahtar Kavram

Writing linear equations from given points and slope, and solving a system of linear equations to identify intercepts.

Tahmini Süre:2m 0s

Soru 31Soru →

For the linear function $f$ , the table below displays selected values of $x$ and their corresponding function values $f(x)$ .

$x$	$f(x)$
$-2$	$1$
$2$	$9$
$4$	$13$

What is the value of $f(10)$ ?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The value of

f(10)

is 25.

The correct answer is 25. By using the points

(-2, 1)

and

(2, 9)

, the slope of the linear function is calculated as

m = \frac{9 - 1}{2 - (-2)} = 2

. Substituting this slope and the point

(2, 9)

into the slope-intercept form,

f(x) = 2x + b

, gives

9 = 2(2) + b

, which resolves to

b = 5

. Thus, the linear function is defined by

f(x) = 2x + 5

. Evaluating the function at

x = 10

yields

f(10) = 2(10) + 5 = 25

.

Adım Adım Çözüm

1

Calculate the slope of the linear function using two coordinate pairs.

Slope

m = 2

The slope of a linear function can be determined by the formula

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

. Substituting

(-2, 1)

and

(2, 9)

yields

m = \frac{9 - 1}{2 - (-2)} = \frac{8}{4} = 2

.

2

Find the y-intercept of the function to write its equation.

f(x) = 2x + 5

Substituting the slope

m = 2

and the point

(2, 9)

into the slope-intercept form

f(x) = mx + b

gives

9 = 2(2) + b

, which simplifies to

b = 5

.

3

Evaluate the function for the input value 10.

f(10) = 25

Substituting

x = 10

into the linear function equation

f(x) = 2x + 5

yields

f(10) = 2(10) + 5 = 25

.

Anahtar Kavram

Determining a linear function from a table of values and using it to find specific outputs.

Tahmini Süre:1m 30s

Soru 32Soru →

A line in the $xy$ -plane is defined by the equation $5y - 2x = 15$ . What is the slope of the line?

\frac{5}{2}

-\frac{2}{5}

\frac{2}{5}

3

Cevabı ve açıklamayı göster

Cevap: $\frac{2}{5}$

Cevap

The slope of the line is

\frac{2}{5}

.

To find the slope of the line, convert the equation

5y - 2x = 15

into slope-intercept form,

y = mx + b

, where

m

represents the slope. First, add

2x

to both sides to get

5y = 2x + 15

. Next, divide both sides by

5

to isolate

y

, yielding

y = \frac{2}{5}x + 3

. In this form, the coefficient of

x

is the slope, which is

\frac{2}{5}

.

Adım Adım Çözüm

1

Write the given equation of the line.

5y - 2x = 15

To identify the slope, we need to manipulate the given equation.

2

Isolate the

y

term on one side by adding

2x

to both sides.

5y = 2x + 15

This is the first step in converting the equation to slope-intercept form (

y = mx + b

).

3

Divide both sides of the equation by

5

to solve for

y

.

y = \frac{2}{5}x + 3

Dividing isolates

y

completely, putting the equation in the form

y = mx + b

, where

m

is the slope and

b

is the

y

-intercept.

4

Identify the slope

m

from the equation.

The slope

m

is

\frac{2}{5}

.

In the equation

y = \frac{2}{5}x + 3

, the coefficient of

x

represents the slope of the line.

Anahtar Kavram

Slope of a line from its equation

Soru 33Soru →

In the $xy$ -plane, the graph of the linear function $f$ passes through the point $(6, 4)$ . Line $g$ is perpendicular to the graph of $f$ and has a $y$ -intercept of $(0, 12)$ . The graph of $f$ and line $g$ intersect at the point $P(h, k)$ , where $h$ and $k$ are integers. If the slope of the graph of $f$ is a positive integer greater than $5$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The value of

k

is

11

.

By equating the linear equations representing

f

and

g

, we find that the intersection point's

x

-coordinate

h

is given by

h = 6 + \frac{8m - 6}{m^2 + 1}

. Since

h

is an integer and

m

is a positive integer greater than

5

, the inequality

m^2 + 1 \le 8m - 6

limits

m

to the range

1 \le m \le 7

. Testing the integers greater than

5

(which are

6

and

7

) reveals that only

m = 7

yields an integer value for

h

, resulting in

h = 7

. Substituting these values back into the equation of either line yields

k = 11

.

Adım Adım Çözüm

1

Write the equation of the linear function

f

in terms of its slope

m

.

y - 4 = m(x - 6) \implies y = mx + 4 - 6m

Since the graph of

f

passes through the point

(6, 4)

with slope

m

, we can use the point-slope form of a linear equation.

2

Write the equation of line

g

in slope-intercept form.

y = -\frac{1}{m}x + 12

Line

g

is perpendicular to the graph of

f

, so its slope is the negative reciprocal of

m

, which is

-\frac{1}{m}

. Its

y

-intercept is

(0, 12)

.

3

Equate the two equations to find the

x

-coordinate

h

of the intersection point

P(h, k)

in terms of

m

.

h = \frac{6m^2 + 8m}{m^2 + 1}

At the intersection point, the

y

-values of both equations are equal, so we solve

mx + 4 - 6m = -\frac{1}{m}x + 12

for

x

.

4

Analyze the expression for

h

to find the integer constraints on

m

.

h = 6 + \frac{8m - 6}{m^2 + 1} \implies m^2 - 8m + 7 \le 0 \implies 1 \le m \le 7

For

h

to be an integer,

\frac{8m - 6}{m^2 + 1}

must be an integer. Since

m

is a positive integer greater than

5

, the numerator is positive, meaning the denominator

m^2 + 1

must be less than or equal to the numerator

8m - 6

.

5

Test the possible integer values of

m

greater than

5

to find the one that yields integer coordinates.

m = 7 \implies h = 7

The only integers greater than

5

in the range

[1, 7]

are

6

and

7

. Testing

m=6

gives a non-integer, while

m=7

gives

h = 7

, which is an integer.

6

Calculate the

y

-coordinate

k

of the intersection point

P

.

k = 11

Substitute

h = 7

into the equation of line

g

:

k = -\frac{1}{7}(7) + 12 = 11

.

Anahtar Kavram

Linear Functions and Graphs

Soru 34Soru →

In the $xy$ -plane, the graph of the linear function $f$ has a slope of $m$ and a $y$ -intercept of $(0, b)$ . The graph of the linear function $g$ has a slope of $-3m$ and a $y$ -intercept of $(0, b + 15)$ . If the graphs of $f$ and $g$ intersect at the point $(3, 5)$ , what is the value of $f(5)$ ?

Cevabı ve açıklamayı göster

Cevap: 7.5

Cevap

7.5 (or 15/2)

By writing the equations of the two lines as

f(x) = mx + b

and

g(x) = -3mx + b + 15

, and substituting the intersection point

(3, 5)

, we get the system of equations

3m + b = 5

and

-9m + b = -10

. Subtracting these equations gives

12m = 15

, which simplifies to

m = 1.25

. Substituting

m

back into

3m + b = 5

yields

b = 1.25

. The function is therefore

f(x) = 1.25x + 1.25

. Evaluating this at

x = 5

gives

f(5) = 1.25(5) + 1.25 = 7.5

.

Adım Adım Çözüm

1

Formulate equations for both linear functions using slope-intercept form.

f(x) = mx + b

and

g(x) = -3mx + b + 15

The equation of a line with slope

M

and

y

-intercept

(0, B)

is given by

y = Mx + B

.

2

Substitute the coordinates of the intersection point

(3, 5)

into both equations.

3m + b = 5

and

-9m + b = -10

Since the intersection point lies on both lines, its coordinates must satisfy both equations.

3

Solve the system of equations for

m

and

b

.

m = 1.25

and

b = 1.25

Subtracting the equations eliminates

b

, giving

12m = 15 \implies m = 1.25

. Substituting this back into the first equation gives

3(1.25) + b = 5 \implies b = 1.25

.

4

Evaluate

f(5)

using the found values.

f(5) = 1.25(5) + 1.25 = 7.5

Substitute

x = 5

into the equation

f(x) = 1.25x + 1.25

.

Anahtar Kavram

Solving systems of linear equations and evaluating linear functions from graph intersections.

Tahmini Süre:2m 0s

Soru 35Soru →

In the $xy$ -plane, a line passes through the origin and has a slope of $\frac{3}{5}$ . If the point $(k, 9)$ lies on the line, what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

15

A line passing through the origin has a

y

-intercept of

0

. Thus, its equation in slope-intercept form is

y = mx

, where

m

is the slope. Given the slope is

\frac{3}{5}

, the equation is

y = \frac{3}{5}x

. Since the point

(k, 9)

lies on the line, substituting these coordinates yields

9 = \frac{3}{5}k

. Multiplying both sides by

\frac{5}{3}

gives

k = 15

.

Adım Adım Çözüm

1

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

y = \frac{3}{5}x

The line passes through the origin

(0,0)

, so the

y

-intercept is

0

, and the slope is

\frac{3}{5}

.

2

Substitute the coordinates of the point

(k, 9)

into the line's equation.

9 = \frac{3}{5}k

A point lies on a line if its coordinates satisfy the equation of the line.

3

Solve the equation for

k

.

k = 15

Multiply both sides of the equation by

\frac{5}{3}

to isolate

k

.

Anahtar Kavram

Using the slope and a point on a line to determine an unknown coordinate.

Soru 36Soru →

A line in the coordinate plane is defined by the function $f(x) = px + q$ , where $p$ and $q$ are constants. This line contains the point $(6, 2)$ and intersects the vertical axis at a point with a negative $y$ -coordinate. If the region bounded by the line and the two coordinate axes is a right triangle with an area of $25$ , what is the value of $p$ ?

\frac{1}{18}

18

2

50

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2

The correct answer is the option representing the value

2

. Substituting the point

(6, 2)

into the linear function gives

q = 2 - 6p

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

\frac{1}{2} \times |\text{base}| \times |\text{height}| = \frac{q^2}{2p}

. Setting this equal to

25

leads to the quadratic equation

18p^2 - 37p + 2 = 0

. Factoring this equation yields

p = 2

or

p = \frac{1}{18}

. Checking the

y

-intercept condition (

q < 0

) shows that only

p = 2

yields a negative

y

-intercept (

q = -10

), whereas

p = \frac{1}{18}

yields a positive

y

-intercept (

q = \frac{5}{3}

).

Adım Adım Çözüm

1

Use the point

(6, 2)

to express

q

in terms of

p

.

q = 2 - 6p

Since the line contains

(6, 2)

, substituting

x = 6

and

f(x) = 2

into the function

f(x) = px + q

must satisfy the equation.

2

Determine the intercepts of the line on the coordinate axes.

The vertical intercept is

(0, q)

and the horizontal intercept is

\left(-\frac{q}{p}, 0\right)

.

The vertical intercept is found by setting

x = 0

, and the horizontal intercept is found by setting

f(x) = 0

.

3

Express the area of the bounded right triangle and set it equal to

25

.

\frac{q^2}{2p} = 25 \Rightarrow q^2 = 50p

The area of a right triangle with vertices at the origin and the intercepts is

\frac{1}{2} \times |\text{base}| \times |\text{height}| = \frac{1}{2} \left(-\frac{q}{p}\right)(-q) = \frac{q^2}{2p}

.

4

Substitute

q = 2 - 6p

into

q^2 = 50p

and solve the quadratic equation.

18p^2 - 37p + 2 = 0 \Rightarrow (18p - 1)(p - 2) = 0 \Rightarrow p = 2

or

p = \frac{1}{18}

.

Substituting the expression for

q

gives a quadratic equation in terms of

p

that can be factored to find its roots.

5

Verify which root satisfies the condition that the vertical intersection point has a negative

y

-coordinate.

For

p = 2

,

q = -10

(valid). For

p = \frac{1}{18}

,

q = \frac{5}{3}

(invalid). Thus,

p = 2

.

The problem states that the vertical axis intersection (

y

-intercept) must have a negative

y

-coordinate (

q < 0

).

Anahtar Kavram

Using coordinate points and geometric properties of intercepts to determine parameters of linear functions

Soru 37Soru →

In the $xy$ -plane, the graph of a linear function $f$ has a negative slope and a positive $y$ -intercept. The graph of $f$ intersects the $x$ -axis at point $A$ and the $y$ -axis at point $B$ . A second line, $L$ , is perpendicular to the graph of $f$ and passes through the origin. If line $L$ intersects the graph of $f$ at point $P$ such that the ratio of the area of triangle $OAP$ to the area of triangle $OBP$ is $4:9$ , what is the slope of the graph of $f$ ?

-\frac{3}{2}

-\frac{2}{3}

-\frac{9}{4}

-\frac{4}{9}

Cevabı ve açıklamayı göster

Cevap: $-\frac{3}{2}$

Cevap

The slope of the graph of

f

is

-\frac{3}{2}

.

The correct answer is

-\frac{3}{2}

. By setting up the equations of the lines

f(x) = mx + b

and

y = -\frac{1}{m}x

, we can express the coordinates of the intercepts

A

and

B

, as well as the intersection point

P

, in terms of

m

and

b

. Calculating the areas of triangle

OAP

and triangle

OBP

gives

\text{Area}(OAP) = \frac{1}{2} \left(-\frac{b}{m}\right) y_0

and

\text{Area}(OBP) = \frac{1}{2} b x_0

. Since

P

lies on the perpendicular line,

y_0 = -\frac{1}{m}x_0

. Substituting this into the area ratio yields

\frac{\text{Area}(OAP)}{\text{Area}(OBP)} = \frac{1}{m^2}

. Equating this to the given ratio

\frac{4}{9}

results in

m^2 = \frac{9}{4}

. Given that the slope is negative,

m

must be

-\frac{3}{2}

.

Adım Adım Çözüm

1

Define the linear function

f(x)

and the line

L

perpendicular to it.

Let the function be

f(x) = mx + b

, where

m < 0

and

b > 0

. The line

L

perpendicular to the graph of

f

that passes through the origin has the equation

y = -\frac{1}{m}x

.

This establishes the equations of both lines in terms of the slope

m

and

y

-intercept

b

.

2

Determine the coordinates of the intercepts

A

and

B

, and the intersection point

P(x_0, y_0)

.

The

x

-intercept is

A\left(-\frac{b}{m}, 0\right)

and the

y

-intercept is

B(0, b)

. The intersection point

P(x_0, y_0)

is the solution to

mx + b = -\frac{1}{m}x

, which gives

x_0 = -\frac{mb}{m^2+1}

and

y_0 = \frac{b}{m^2+1}

.

Finding these points allows us to express the dimensions of triangles

OAP

and

OBP

.

3

Calculate the areas of triangles

OAP

and

OBP

and find their ratio in terms of

m

.

The base of triangle

OAP

along the

x

-axis is

OA = -\frac{b}{m}

and its height is

y_0

. Thus,

\text{Area}(OAP) = \frac{1}{2} \left(-\frac{b}{m}\right) y_0

. The base of triangle

OBP

along the

y

-axis is

OB = b

and its height is

x_0

. Thus,

\text{Area}(OBP) = \frac{1}{2} b x_0

. The ratio of their areas is

\frac{\text{Area}(OAP)}{\text{Area}(OBP)} = \frac{-\frac{b}{m} y_0}{b x_0} = -\frac{y_0}{m x_0}

. Since

P

lies on

L

, we have

\frac{y_0}{x_0} = -\frac{1}{m}

. Substituting this gives the ratio

\frac{\text{Area}(OAP)}{\text{Area}(OBP)} = -\frac{1}{m}\left(-\frac{1}{m}\right) = \frac{1}{m^2}

.

This simplifies the geometric relationship to a direct relation between the area ratio and the slope of the function.

4

Solve for the slope

m

using the given ratio of

4:9

.

Setting

\frac{1}{m^2} = \frac{4}{9}

yields

m^2 = \frac{9}{4}

. Since the problem states the slope is negative, we take the negative square root to get

m = -\frac{3}{2}

.

This identifies the correct slope value matching the given constraints.

Anahtar Kavram

Using coordinate geometry and system of linear equations to determine slopes and intersections, and relating those to geometric areas on the coordinate plane.

Soru 38Soru →

For the linear function $f$ , the value of $f(0)$ is $3$ and the value of $f(5)$ is $18$ . What is the slope of the graph of $y = f(x)$ in the $xy$ -plane?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The slope of the graph of

y = f(x)

is

3

.

The slope of a linear function can be determined using any two points

(x_1, y_1)

and

(x_2, y_2)

on its graph with the formula

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

. The given function values

f(0) = 3

and

f(5) = 18

correspond to the coordinate points

(0, 3)

and

(5, 18)

respectively. Substituting these coordinates into the formula gives

m = \frac{18 - 3}{5 - 0} = \frac{15}{5} = 3

.

Adım Adım Çözüm

1

Identify the coordinates of two points on the line using the given function values.

The function values

f(0) = 3

and

f(5) = 18

correspond to the points

(0, 3)

and

(5, 18)

on the graph of the function.

In function notation,

f(x) = y

represents a point

(x, y)

on the graph of the function.

2

Calculate the slope using the slope formula.

The slope

m

is calculated as

\frac{18 - 3}{5 - 0} = 3

.

The slope of a line is defined as the change in

y

divided by the change in

x

between any two points on the line.

Anahtar Kavram

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

Soru 39Soru →

A linear function $f$ is defined by $f(x) = mx + b$ , where $m$ and $b$ are constants. The graph of $f$ in the $xy$ -plane passes through the point $(2, 5)$ . A second linear function $g$ is defined by $g(x) = f(x - 3) + 4$ . The graph of $g$ has an $x$ -intercept of $(k, 0)$ and a $y$ -intercept of $(0, -k)$ , where $k$ is a nonzero constant. What is the value of $f(8)$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

11

The correct answer is 11. By using the given point on the graph of

f(x)

, we express its y-intercept

b

in terms of its slope

m

as

b = 5 - 2m

. Substituting this expression into the translation equation

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

yields

g(x) = mx - 5m + 9

. Setting the y-intercept of

g(x)

, which is

-5m + 9

, equal to

-k

gives

k = 5m - 9

. Evaluating

g(x)

at its x-intercept

x = k

gives

mk - 5m + 9 = 0

. Substituting the relation

-5m + 9 = -k

into this equation yields

mk - k = 0

, or

k(m - 1) = 0

. Since

k

is nonzero, we find

m = 1

, which in turn gives

b = 3

. Thus, the linear function is

f(x) = x + 3

, and

f(8) = 11

.

Adım Adım Çözüm

1

Express the y-intercept constant

b

in terms of the slope

m

.

b = 5 - 2m

Since the graph of

f(x) = mx + b

passes through

(2, 5)

, substituting

x = 2

and

f(x) = 5

gives

5 = 2m + b

.

2

Write the expression for

g(x)

in terms of

m

.

g(x) = mx - 5m + 9

By definition,

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

. Substituting

b = 5 - 2m

simplifies the expression to

g(x) = mx - 5m + 9

.

3

Relate the y-intercept of

g(x)

to the parameter

k

.

k = 5m - 9

The y-intercept of

g(x)

is

g(0) = -5m + 9

. Since the y-intercept is given as

(0, -k)

, we set

-k = -5m + 9

, which gives

k = 5m - 9

.

4

Set up an equation using the x-intercept of

g(x)

.

g(k) = mk - 5m + 9 = 0

Since the x-intercept of

g(x)

is

(k, 0)

, substituting

x = k

into the expression for

g(x)

must yield

0

.

5

Solve for the slope

m

.

m = 1

Substituting

-5m + 9 = -k

into the equation

mk - 5m + 9 = 0

gives

mk - k = 0

, which factors as

k(m - 1) = 0

. Since

k

is a nonzero constant, we divide by

k

to get

m - 1 = 0

, so

m = 1

.

6

Find the constant

b

and write the final formula for

f(x)

.

f(x) = x + 3

Substituting

m = 1

back into

b = 5 - 2m

yields

b = 3

. Therefore,

f(x) = x + 3

.

7

Evaluate

f(8)

.

11

Substituting

x = 8

into the function

f(x) = x + 3

gives

f(8) = 8 + 3 = 11

.

Anahtar Kavram

Linear function transformations and intercept properties

Soru 40Soru →

The graph of the linear function $f$ in the $xy$ -plane is defined by $f(x) = kx + 4$ , where $k$ is a positive constant. The graph of the linear function $g$ is obtained by translating the graph of $f$ left by $3$ units and down by $2$ units. If the $x$ -intercept of the graph of $g$ is $5$ times the $x$ -intercept of the graph of $f$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

The correct answer is 6. The linear function

g(x)

represents the translated graph, which is expressed as

g(x) = f(x+3) - 2

. Substituting

f(x) = kx + 4

yields

g(x) = k(x+3) + 4 - 2 = kx + 3k + 2

. The

x

-intercept of a graph is the point where the function value is

0

. Solving

f(x) = 0

gives the

x

-intercept of the graph of

f

as

x = -\frac{4}{k}

. Solving

g(x) = 0

gives the

x

-intercept of the graph of

g

as

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

. We are given that the

x

-intercept of the graph of

g

is

5

times the

x

-intercept of the graph of

f

, which translates to the equation

-\frac{3k+2}{k} = 5 \left(-\frac{4}{k}\right)

. Multiplying both sides by

-k

(since

k > 0

,

k \neq 0

) yields

3k + 2 = 20

. Subtracting

2

from both sides gives

3k = 18

, and dividing by

3

results in

k = 6

.

Adım Adım Çözüm

1

Determine the equation of the translated function

g(x)

g(x) = kx + 3k + 2

Shifting the graph of

f(x)

left by

3

units and down by

2

units translates to

g(x) = f(x+3) - 2

.

2

Find the

x

-intercept of the graph of

f

x = -\frac{4}{k}

The

x

-intercept is the input value for which the output of the function is zero, so we solve

f(x) = 0

.

3

Find the

x

-intercept of the graph of

g

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

We set the equation of

g(x)

to zero and solve for

x

in terms of

k

.

4

Set up and solve the equation relating the two

x

-intercepts

k = 6

Using the relation

x_g = 5 \cdot x_f

, we obtain

-\frac{3k + 2}{k} = 5 \left(-\frac{4}{k}\right)

, which simplifies to

3k + 2 = 20

, leading to

k = 6

.

Anahtar Kavram

Function translations and finding intercepts of linear equations

Linear Functions and Graphs

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