Advanced Math Practice Questions — SAT — Page 20

Question 381Question →

A transformation is applied to the exponential function $f(x) = 2^x - 3$ to obtain a new function $g$ , where $g(x) = -f(x - 1) + 2$ . What is the $y$ -intercept of the graph of $y = g(x)$ in the $xy$ -plane?

(0, 3)

(0, -\frac{3}{2})

(0, \frac{9}{2})

(0, -\frac{1}{2})

Show answer & explanation

Answer: $(0, \frac{9}{2})$

Answer

(0, \frac{9}{2})

The correct answer is the coordinate point with

y

-coordinate

\frac{9}{2}

. The

y

-intercept of the graph of

g(x)

is found by evaluating

g(0)

. Substituting

x = 0

into the transformation equation gives

g(0) = -f(-1) + 2

. Using the definition of

f(x)

, we find

f(-1) = 2^{-1} - 3 = 0.5 - 3 = -2.5

. Substituting this value back into the equation for

g(0)

yields

g(0) = -(-2.5) + 2 = 2.5 + 2 = 4.5

, which is equivalent to

\frac{9}{2}

. Therefore, the

y

-intercept is the coordinate point containing

\frac{9}{2}

.

Step-by-Step Solution

1

Substitute

x = 0

into the definition of

g(x)

to find the

y

-coordinate of the

y

-intercept.

g(0) = -f(0 - 1) + 2 = -f(-1) + 2

The

y

-intercept of any function

y = g(x)

occurs where the input

x

is equal to

0

.

2

Evaluate

f(-1)

using the definition

f(x) = 2^x - 3

.

f(-1) = 2^{-1} - 3 = \frac{1}{2} - 3 = -2.5

Evaluating

f(-1)

is necessary to substitute its value back into the expression for

g(0)

.

3

Substitute the value of

f(-1)

back into the expression for

g(0)

and simplify.

g(0) = -(-2.5) + 2 = 2.5 + 2 = 4.5 = \frac{9}{2}

This yields the final

y

-coordinate of the

y

-intercept, which is

\frac{9}{2}

, giving the coordinate point

(0, \frac{9}{2})

.

Key Concept

Applying horizontal translation, reflection across the x-axis, and vertical translation to evaluate a function at a specific point.

Question 382Question →

The height of a diver, in meters, above the pool surface $t$ seconds after leaving the diving board is modeled by the function $f(t) = -4.9(t - 1)^2 + 10$ . If the diving board is moved so that the diver's trajectory is shifted $0.5$ seconds later in time and the maximum height is increased by $2$ meters, which of the following functions $g$ models the diver's new trajectory?

g(t) = -4.9(t - 0.5)^2 + 12

g(t) = -4.9(t - 1.5)^2 + 8

g(t) = -4.9(t - 1.5)^2 + 12

g(t) = -4.9(t + 0.5)^2 + 12

Show answer & explanation

Answer: $g(t) = -4.9(t - 1.5)^2 + 12$

Answer

The function

g(t) = -4.9(t - 1.5)^2 + 12

models the diver's new trajectory.

The correct answer represents the translated quadratic function. Since the original function

f(t) = -4.9(t - 1)^2 + 10

has its vertex at

(1, 10)

, shifting the trajectory

0.5

seconds later in time moves the vertex horizontally to the right to

t = 1.5

seconds, replacing

(t - 1)

with

(t - 1.5)

. Increasing the maximum height by

2

meters shifts the vertex vertically upward to

12

meters, replacing the constant term

10

with

12

. This results in the function

g(t) = -4.9(t - 1.5)^2 + 12

.

Step-by-Step Solution

1

Identify the vertex and its meaning in the original function

f(t) = -4.9(t - 1)^2 + 10

.

The vertex is

(1, 10)

, indicating that the maximum height of

10

meters occurs at

t = 1

second.

The vertex form of a quadratic function is

y = a(t - h)^2 + k

, where

(h, k)

is the vertex representing the extreme value.

2

Apply the horizontal translation of

0.5

seconds later in time.

The new

t

-coordinate of the vertex is

1 + 0.5 = 1.5

seconds, which changes the term

(t - 1)

to

(t - 1.5)

.

A horizontal shift to the right by

c

units is represented by replacing the variable

t

with

t - c

.

3

Apply the vertical translation of

2

meters upward.

The new

y

-coordinate of the vertex is

10 + 2 = 12

meters, which changes the constant term of the function to

12

.

A vertical shift upward by

d

units is represented by adding

d

to the function value.

4

Combine the horizontal and vertical translations to write the new equation.

g(t) = -4.9(t - 1.5)^2 + 12

.

Substituting the shifted vertex coordinates

(1.5, 12)

into the vertex form while keeping the same vertical stretch/direction factor

a = -4.9

yields the final equation.

Key Concept

Vertex form and transformations of quadratic functions

Question 383Question →

The function $f$ is defined by $f(x) = 2x^2 - 12x + c$ , where $c$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ has a vertex at $(h, 5)$ , where $h$ is a constant. What is the value of $c$ ?

Show answer & explanation

Answer: 23

Answer

23

For a quadratic function in standard form

f(x) = ax^2 + bx + c

, the x-coordinate of the vertex is given by

h = -\frac{b}{2a}

. In this function,

a = 2

and

b = -12

, so

h = -\frac{-12}{2(2)} = 3

. The vertex coordinates are

(3, 5)

, which means

f(3) = 5

. Substituting

x = 3

into the function gives

2(3)^2 - 12(3) + c = 5

. Simplifying the terms gives

18 - 36 + c = 5

, which becomes

-18 + c = 5

. Adding

18

to both sides yields

c = 23

.

Step-by-Step Solution

1

Identify the x-coordinate formula for the vertex of a quadratic function

f(x) = ax^2 + bx + c

.

The x-coordinate

h

is given by

h = -\frac{b}{2a}

.

To find the axis of symmetry and the horizontal position of the vertex.

2

Substitute

a = 2

and

b = -12

from the given equation

f(x) = 2x^2 - 12x + c

into the vertex formula.

h = -\frac{-12}{2(2)} = 3

To calculate the specific x-coordinate of the vertex for this function.

3

Substitute the vertex coordinates

(3, 5)

into the function

f(x)

.

f(3) = 2(3)^2 - 12(3) + c = 5

Since the vertex lies on the graph of the function, its coordinates must satisfy the function's equation.

4

Simplify the equation and solve for the constant

c

.

18 - 36 + c = 5 \implies -18 + c = 5 \implies c = 23

To determine the final value of the constant

c

.

Key Concept

Vertex of a quadratic function

Question 384Question →

The function $f$ is defined by $f(x) = |x - 2| - 3$ . The function $g$ is defined by $g(x) = -f(x + 1) + 2$ . What is the maximum value of $g(x)$ ?

-1

1

3

5

Show answer & explanation

Answer: 5

Answer

5

The vertex of the absolute value function

f(x) = |x - 2| - 3

is at

(2, -3)

, meaning its minimum value is

-3

. The transformed function

g(x) = -f(x + 1) + 2

shifts the graph 1 unit to the left, reflects it vertically across the

x

-axis (which turns the minimum value of

-3

into a maximum value of

3

), and translates it 2 units upward. This results in a final maximum value of

3 + 2 = 5

.

Step-by-Step Solution

1

Find the minimum value of

f(x)

and the location of its vertex.

The vertex of

f(x) = |x - 2| - 3

is at

(2, -3)

, which means

f(x)

has a minimum value of

-3

at

x = 2

.

Since the coefficient of the absolute value expression is positive, the graph of

f

opens upward, making the vertex a minimum point.

2

Apply the horizontal translation to find the new vertex position.

The term

f(x + 1)

shifts the graph 1 unit to the left, moving the vertex from

x = 2

to

x = 1

.

Adding a constant inside the function argument,

f(x + h)

where

h > 0

, shifts the graph horizontally to the left by

h

units.

3

Apply the vertical reflection to the minimum value.

The term

-f(x + 1)

reflects the graph across the

x

-axis. The vertex point

(1, -3)

becomes

(1, 3)

, and the minimum value of

-3

becomes a maximum value of

3

.

Multiplying the function by

-1

reflects the graph vertically, swapping all positive and negative

y

-values and changing the minimum value to a maximum value.

4

Apply the vertical translation to find the final maximum value.

Adding 2 to the function,

g(x) = -f(x + 1) + 2

, shifts the entire graph vertically upward by 2 units. The vertex moves from

(1, 3)

to

(1, 5)

.

Adding a constant to the outside of the function shifts the graph vertically, so the maximum value increases from 3 to 5.

Key Concept

Function Notation and Transformations

Question 385Question →

The graph of the quadratic function $f(x) = x^2 - 4x + 7$ is translated $3$ units to the right and $2$ units down in the $xy$ -plane to form the graph of the function $g(x) = x^2 + px + q$ , where $p$ and $q$ are constants. What is the value of $q$ ?

Show answer & explanation

Answer: 26

Answer

The value of

q

is

26

.

To find the constant term

q

of the translated quadratic function

g(x) = x^2 + px + q

, we first determine the vertex of the original function

f(x) = x^2 - 4x + 7

. Completing the square gives

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

, which identifies the vertex of the parabola as

(2, 3)

. Translating the graph

3

units to the right and

2

units down shifts the vertex to

(2 + 3, 3 - 2) = (5, 1)

. Because the translation does not affect the shape of the parabola, the leading coefficient remains

1

. The vertex form of the new function is

g(x) = (x - 5)^2 + 1

. Expanding this expression yields

g(x) = x^2 - 10x + 25 + 1 = x^2 - 10x + 26

. Comparing this to

g(x) = x^2 + px + q

, we find that

q = 26

.

Step-by-Step Solution

1

Convert the original function

f(x) = x^2 - 4x + 7

into vertex form,

f(x) = a(x - h)^2 + k

, to identify its vertex

(h, k)

.

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

, which represents a parabola with vertex

(2, 3)

.

Finding the vertex of the original function allows us to apply the translation directly to the vertex coordinates.

2

Apply the translation of

3

units to the right and

2

units down to the coordinates of the vertex

(2, 3)

.

The new vertex is

(2 + 3, 3 - 2) = (5, 1)

.

Translating a graph shifts its vertex by the corresponding horizontal and vertical amounts.

3

Write the equation of the translated function

g(x)

in vertex form using the new vertex

(5, 1)

and the original leading coefficient

a = 1

.

g(x) = (x - 5)^2 + 1

A translation does not change the shape or vertical stretch of the parabola, so the coefficient of

x^2

remains

1

.

4

Expand the vertex form of

g(x)

into standard form,

g(x) = x^2 + px + q

, to determine the constant term

q

.

g(x) = x^2 - 10x + 26

, which means

q = 26

.

Expanding the equation allows us to compare it directly with the standard form of

g(x)

and identify the value of the constant term.

Key Concept

Vertex form and translations of quadratic functions

Alternative Method

Alternatively, the translation can be applied directly to the variable

x

in the function equation. Translating a function

f(x)

by

3

units to the right and

2

units down yields

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

. Substituting

x - 3

into the original function gives:

g(x) = (x - 3)^2 - 4(x - 3) + 7 - 2

. Simplifying this expression:

g(x) = (x^2 - 6x + 9) - (4x - 12) + 5 = x^2 - 10x + 26

. This directly shows that the constant term

q

is

26

.

Estimated Time:1m 30s

Question 386Question →

A company's daily profit, $P(x)$ , in dollars, from selling $x$ units of a product is modeled by the function $P(x) = -2x^2 + kx - 800$ , where $k$ is a constant. If the company achieves its maximum daily profit of $\$1000$ when it sells $30$ units, what is the value of $k$ ?

120

60

-120

180

Show answer & explanation

Answer: 120

Answer

120

The maximum value of a quadratic function occurs at its vertex. Given that the vertex is at

(30, 1000)

and the coefficient of the squared term is

-2

, the profit function can be written in vertex form as

P(x) = -2(x - 30)^2 + 1000

. Expanding this expression yields

P(x) = -2(x^2 - 60x + 900) + 1000 = -2x^2 + 120x - 800

. Comparing this to the given equation

P(x) = -2x^2 + kx - 800

, the coefficient of

x

must be 120. Alternatively, using the vertex formula

h = -b/(2a)

with

h = 30

and

a = -2

gives

30 = -k/(2 \times -2)

, which simplifies to

30 = k/4

and results in

k = 120

.

Step-by-Step Solution

1

Identify the vertex coordinates from the given context.

The vertex of the profit parabola is at

(h, d) = (30, 1000)

.

The maximum profit of

\$1000

occurs when

30

units are sold, representing the peak of the downward-opening parabola.

2

Substitute the vertex and the leading coefficient

a = -2

into the vertex form of a quadratic function,

P(x) = a(x - h)^2 + d

.

The equation becomes

P(x) = -2(x - 30)^2 + 1000

.

The vertex form allows direct substitution of the vertex coordinates to build the function's equation.

3

Expand the vertex form equation into standard form.

P(x) = -2(x^2 - 60x + 900) + 1000 = -2x^2 + 120x - 800

.

Expanding the equation allows direct comparison of terms with the given standard form

P(x) = -2x^2 + kx - 800

.

4

Compare the coefficient of the

x

term in the expanded equation to the coefficient of the

x

term in the given equation.

k = 120

.

Corresponding coefficients of identical functions must be equal, allowing us to determine the value of the constant

k

.

Key Concept

Quadratic Functions and Graphs

Question 387Question →

In the quadratic equation $x^2 - kx + 36 = 0$ , $k$ is a positive constant. If the difference between the two solutions to the equation is $5$ , what is the value of $k$ ?

Show answer & explanation

Answer: 13

Answer

The value of the positive constant

k

is

13

.

For the quadratic equation

x^2 - kx + 36 = 0

with solutions

r_1

and

r_2

, the sum of the solutions is

r_1 + r_2 = k

and the product of the solutions is

r_1 \cdot r_2 = 36

. Given that the difference between the two solutions is

5

, we can write

|r_1 - r_2| = 5

. Squaring both sides yields

(r_1 - r_2)^2 = 25

. Using the algebraic identity

(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2

, we substitute the known values to obtain

25 = k^2 - 4(36)

, which simplifies to

25 = k^2 - 144

. Solving for

k^2

gives

k^2 = 169

. Since

k

is positive,

k = 13

. Alternatively, we can find two numbers whose product is

36

and whose difference is

5

. These numbers are

9

and

4

, because

9 \cdot 4 = 36

and

9 - 4 = 5

. The sum of these solutions is

9 + 4 = 13

, which matches the coefficient of the linear term,

k

.

Step-by-Step Solution

1

Relate the roots of the quadratic equation

x^2 - kx + 36 = 0

to its coefficients using Vieta's formulas.

The sum of the roots is

r_1 + r_2 = k

and the product of the roots is

r_1 \cdot r_2 = 36

.

This sets up the system of equations representing the roots.

2

Express the given root difference of

5

mathematically and square it.

|r_1 - r_2| = 5 \implies (r_1 - r_2)^2 = 25

.

Squaring the difference allows us to use standard algebraic identities.

3

Apply the identity

(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2

to substitute the sum and product expressions.

25 = k^2 - 4(36)

This converts the relationship between the roots into an equation with the single variable

k

.

4

Solve the equation for

k

, selecting the positive value.

25 = k^2 - 144 \implies k^2 = 169 \implies k = 13

(since

k > 0

).

To determine the final value of

k

satisfying the constraint that

k

is a positive constant.

Key Concept

Relationship between the roots and coefficients of a quadratic equation (Vieta's formulas)

Question 388Question →

A quadratic function $f$ has its vertex at $(4, 12)$ and a $y$ -intercept at $(0, -4)$ in the $xy$ -plane. The function $g$ is defined by $g(x) = f(x + 2) + k$ , where $k$ is a constant. If the $y$ -intercept of the graph of $g$ is $(0, 15)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 7

Answer

7

The quadratic function

f(x)

is determined to be

f(x) = -(x - 4)^2 + 12

by substituting the vertex

(4, 12)

and the

y

-intercept

(0, -4)

into the vertex form. The transformation

g(x) = f(x + 2) + k

translates the function horizontally left by 2 units and vertically by

k

units, resulting in

g(x) = -(x - 2)^2 + 12 + k

. Using the

y

-intercept of

g

, which is

(0, 15)

, we substitute

x = 0

to get

15 = -(0 - 2)^2 + 12 + k

, simplifying to

15 = 8 + k

, which yields

k = 7

.

Step-by-Step Solution

1

Write the vertex form of

f(x)

f(x) = a(x - 4)^2 + 12

The vertex form of a quadratic function with vertex

(h, k_{vertex})

is given by

f(x) = a(x - h)^2 + k_{vertex}

.

2

Determine the value of the coefficient

a

a = -1

, so

f(x) = -(x - 4)^2 + 12

Substitute the coordinates of the

y

-intercept

(0, -4)

into the vertex form equation to solve for

a

.

3

Express the transformed function

g(x)

in terms of

x

and

k

g(x) = -(x - 2)^2 + 12 + k

Apply the translation rules: substituting

x + 2

for

x

shifts the graph left by 2 units, and adding

k

shifts the graph vertically by

k

units.

4

Solve for the constant

k

k = 7

Use the

y

-intercept of

g(x)

, which is

(0, 15)

, so

g(0) = 15

. Setting

15 = -(0 - 2)^2 + 12 + k

simplifies to

15 = 8 + k

, which yields

k = 7

.

Key Concept

Quadratic function vertex form and transformations

Question 389Question →

The polynomial function $f$ is defined by $f(x) = x^3 - 2x^2 - 13x + k$ , where $k$ is a constant. If $x - 4$ is a factor of $f(x)$ , what is the value of $f(-2)$ ?

-22

30

46

54

Show answer & explanation

Answer: $30$

Answer

The correct value of

f(-2)

is

30

.

The correct answer is

30

. According to the Factor Theorem, if

x - 4

is a factor of

f(x)

, then

f(4) = 0

. Substituting

x = 4

into

f(x) = x^3 - 2x^2 - 13x + k

gives

4^3 - 2(4)^2 - 13(4) + k = 0

. Simplifying this expression results in

64 - 32 - 52 + k = 0

, which gives

-20 + k = 0

, so

k = 20

. The polynomial is therefore

f(x) = x^3 - 2x^2 - 13x + 20

. Evaluating this function at

x = -2

gives

f(-2) = (-2)^3 - 2(-2)^2 - 13(-2) + 20 = -8 - 8 + 26 + 20 = 30

.

Step-by-Step Solution

1

Apply the Factor Theorem to set up an equation for

k

.

f(4) = 0

According to the Factor Theorem, if

x - c

is a factor of a polynomial

f(x)

, then

f(c) = 0

. Since

x - 4

is a factor, substituting

x = 4

into

f(x)

must yield

0

.

2

Substitute

x = 4

into the polynomial and solve for

k

.

k = 20

Evaluating

f(4)

gives

4^3 - 2(4)^2 - 13(4) + k = 0

. This simplifies to

64 - 32 - 52 + k = 0

, which simplifies further to

-20 + k = 0

. Solving this equation gives

k = 20

.

3

Write the complete polynomial function using the solved value of

k

.

f(x) = x^3 - 2x^2 - 13x + 20

Replacing the constant

k

with

20

in the original function definition gives the complete polynomial expression.

4

Evaluate the polynomial function at

x = -2

.

f(-2) = 30

Substituting

x = -2

into the polynomial gives

f(-2) = (-2)^3 - 2(-2)^2 - 13(-2) + 20

. Evaluating the terms gives

-8 - 8 + 26 + 20 = 30

.

Key Concept

The Factor Theorem states that a polynomial

f(x)

has a factor

x - c

if and only if

f(c) = 0

. This can be used to determine unknown coefficients in a polynomial before evaluating the function at a different value.

Question 390Question →

In the $xy$ -plane, the graph of the quadratic function $f$ is a parabola with vertex $(4, -3)$ . If the graph passes through the point $(1, 15)$ , what is the value of $f(2)$ ?

Show answer & explanation

Answer: 5

Answer

The value of

f(2)

is

5

.

By using the vertex form of a quadratic function,

f(x) = a(x - h)^2 + k

with vertex

(4, -3)

, the function can be written as

f(x) = a(x - 4)^2 - 3

. Substituting the point

(1, 15)

gives

15 = a(1 - 4)^2 - 3

, which simplifies to

18 = 9a

, leading to

a = 2

. Substituting

a = 2

back into the function gives

f(x) = 2(x - 4)^2 - 3

. Finally, evaluating at

x = 2

gives

f(2) = 2(2 - 4)^2 - 3 = 5

.

Step-by-Step Solution

1

Write the quadratic function in vertex form and substitute the vertex

(4, -3)

.

f(x) = a(x - 4)^2 - 3

The vertex form of a quadratic function is

f(x) = a(x - h)^2 + k

, where

(h, k)

is the vertex.

2

Substitute the point

(1, 15)

into the equation and solve for the constant

a

.

a = 2

Since the graph passes through

(1, 15)

, substituting

x = 1

and

f(x) = 15

allows us to solve for

a

.

3

Substitute

x = 2

into the completed function

f(x) = 2(x - 4)^2 - 3

to find

f(2)

.

f(2) = 5

Evaluating the function at

x = 2

yields the required value.

Key Concept

Determining a quadratic function's equation from its vertex and a point, then evaluating it.

Question 391Question →

A polynomial function $f$ has the form $f(x) = a(x - 2)(x + 3)(x - 5)$ , where $a$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ has a $y$ -intercept of $(0, 60)$ . What is the value of $f(1)$ ?

Show answer & explanation

Answer: 32

Answer

32

The

y

-intercept of the graph is given as

(0, 60)

, which means that when

x = 0

,

f(0) = 60

. Substituting

x = 0

into the function gives

f(0) = a(0 - 2)(0 + 3)(0 - 5) = 30a

. Equating this to 60 gives

30a = 60

, so

a = 2

. Therefore, the function is

f(x) = 2(x - 2)(x + 3)(x - 5)

. To find the value of

f(1)

, substitute

x = 1

into this expression:

f(1) = 2(1 - 2)(1 + 3)(1 - 5) = 2(-1)(4)(-4) = 32

.

Step-by-Step Solution

1

Identify the relation between the

y

-intercept and the function's value.

f(0) = 60

The

y

-intercept of a graph

y = f(x)

is the point where

x = 0

.

2

Substitute

x = 0

into the definition of

f(x)

and set it equal to 60.

a(0 - 2)(0 + 3)(0 - 5) = 60 \implies 30a = 60

This allows us to solve for the unknown constant coefficient

a

.

3

Solve the linear equation for

a

.

a = 2

Dividing both sides of the equation by 30 isolates

a

.

4

Evaluate the complete function

f(x) = 2(x - 2)(x + 3)(x - 5)

at

x = 1

.

f(1) = 2(1 - 2)(1 + 3)(1 - 5) = 2(-1)(4)(-4) = 32

This yields the requested value of

f(1)

.

Key Concept

Using the factors and a known point (such as the y-intercept) of a polynomial function to determine its algebraic expression and evaluate it.

Question 392Question →

The tables below show some values of the function $f$ and the transformed function $g$ , where $g(x) = f(x - h) + k$ for constants $h$ and $k$ .

$x$	$f(x)$
$-2$	$8$
$0$	$3$
$2$	$-1$
$4$	$5$

$x$	$g(x)$
$1$	$6$
$3$	$1$
$5$	$-3$
$7$	$3$

What is the value of $h + k$ ?

5

1

-5

-1

Show answer & explanation

Answer: 1

Answer

1

The correct answer is 1. By comparing the tables, each input of

g(x)

is

3

units greater than the corresponding input of

f(x)

(for example,

g(1)

corresponds to

f(-2)

because

1 - (-2) = 3

). A horizontal shift of

3

units to the right is represented in function notation by subtracting

3

from the input variable, so

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

, which gives

h = 3

. Next, comparing the output values shows that each output of

g(x)

is

2

units less than the corresponding output of

f(x)

(for example,

g(1) = 6

while

f(-2) = 8

, and

6 = 8 - 2

). This indicates a vertical shift downwards by

2

units, so

k = -2

. Therefore, the value of

h + k

is

3 + (-2) = 1

.

Step-by-Step Solution

1

Determine the horizontal translation constant

h

by comparing the inputs of

f(x)

and

g(x)

that correspond to related output values.

The inputs for

f(x)

are

\{-2, 0, 2, 4\}

and the corresponding inputs for

g(x)

are

\{1, 3, 5, 7\}

. Each input for

g(x)

is

3

units greater than the corresponding input for

f(x)

(since

1 - (-2) = 3

,

3 - 0 = 3

, etc.). This indicates a horizontal shift of

3

units to the right, which means the argument of

f

in

g(x)

must be

x - 3

. Comparing this with

f(x - h)

gives

h = 3

.

Identifying the horizontal shift determines the value of the parameter

h

in the translation equation

g(x) = f(x - h) + k

.

2

Determine the vertical translation constant

k

by comparing the output values of

g(x)

with the corresponding values of

f(x - 3)

.

Using the matching inputs, compare the outputs:
- For

x = 1

,

g(1) = 6

and

f(1 - 3) = f(-2) = 8

. The difference is

6 - 8 = -2

.
- For

x = 3

,

g(3) = 1

and

f(3 - 3) = f(0) = 3

. The difference is

1 - 3 = -2

.
- For

x = 5

,

g(5) = -3

and

f(5 - 3) = f(2) = -1

. The difference is

-3 - (-1) = -2

.
- For

x = 7

,

g(7) = 3

and

f(7 - 3) = f(4) = 5

. The difference is

3 - 5 = -2

.
Since each output of

g(x)

is

2

units less than the corresponding output of

f(x - 3)

, the vertical translation is

k = -2

.

Comparing the corresponding outputs identifies the vertical shift constant

k

in the translation equation

g(x) = f(x - h) + k

.

3

Calculate the sum of the constants

h

and

k

.

h + k = 3 + (-2) = 1

.

Evaluating the sum of the parameters provides the final value requested by the question.

Key Concept

Function Notation and Transformations

Alternative Method

Instead of analyzing the general shift for all points, you can choose a single corresponding pair of points from the tables to set up equations. Let the point

(-2, 8)

from the table of

f

correspond to the point

(1, 6)

from the table of

g

. Since

g(x) = f(x - h) + k

, we substitute

x = 1

to get

g(1) = f(1 - h) + k

. Substituting the values

g(1) = 6

and setting the input of

f

to match, we get

1 - h = -2 \implies h = 3

. This simplifies the equation to

6 = f(-2) + k \implies 6 = 8 + k \implies k = -2

. This gives

h + k = 3 - 2 = 1

.

Estimated Time:1m 30s

Question 393Question →

The trajectory of a basketball thrown toward a hoop can be modeled by a quadratic function. In the $xy$ -plane, $x$ represents the horizontal distance in feet from the shooter and $y$ represents the height of the basketball in feet. The basketball reaches its maximum height of $14$ feet at a horizontal distance of $8$ feet from the shooter. If the basketball is released at a height of $6$ feet, which of the following equations models the trajectory of the basketball?

y = -\frac{1}{8}(x + 8)^2 + 14

y = -\frac{1}{8}(x - 14)^2 + 8

y = -\frac{1}{8}(x - 8)^2 + 14

y = -8(x - 8)^2 + 14

Show answer & explanation

Answer: $y = -\frac{1}{8}(x - 8)^2 + 14$

Answer

The equation

y = -\frac{1}{8}(x - 8)^2 + 14

The vertex form of a quadratic function is

y = a(x - h)^2 + k

, where

(h, k)

represents the coordinates of the vertex. Since the maximum height of the basketball is

14

feet at a horizontal distance of

8

feet, the vertex is

(8, 14)

. Substituting these values gives the equation

y = a(x - 8)^2 + 14

. The release point represents the y-intercept where

x = 0

and

y = 6

. Substituting these coordinates into the equation gives

6 = a(0 - 8)^2 + 14

, which simplifies to

6 = 64a + 14

. Subtracting

14

from both sides results in

-8 = 64a

, and dividing both sides by

64

yields

a = -\frac{1}{8}

. Therefore, the correct trajectory model is

y = -\frac{1}{8}(x - 8)^2 + 14

.

Step-by-Step Solution

1

Identify the vertex coordinates

(h, k)

of the parabolic path from the given maximum height context.

The vertex

(h, k)

is

(8, 14)

.

The basketball reaches its maximum height of

14

feet at a horizontal distance of

8

feet, and the maximum of a downward-opening parabola is its vertex.

2

Write the general vertex form of a quadratic equation and substitute the coordinates of the vertex.

y = a(x - 8)^2 + 14

The vertex form of a quadratic equation is

y = a(x - h)^2 + k

, where

(h, k)

is the vertex.

3

Substitute the initial release point

(0, 6)

into the equation to solve for the coefficient

a

.

6 = a(0 - 8)^2 + 14 \implies 6 = 64a + 14 \implies -8 = 64a \implies a = -\frac{8}{64} = -\frac{1}{8}

The basketball is released at a height of

6

feet when the horizontal distance

x = 0

, representing the y-intercept of the trajectory.

4

Substitute the solved value of

a

back into the vertex form equation.

y = -\frac{1}{8}(x - 8)^2 + 14

This yields the complete quadratic model representing the path of the basketball.

Key Concept

Writing quadratic equations in vertex form from context

Question 394Question →

In the $xy$ -plane, a cubic polynomial function $f$ crosses the $x$ -axis at $(-3, 0)$ , $(1, 0)$ , and $(4, 0)$ . Given that $f(2) = -10$ , what is the $y$ -coordinate of the $y$ -intercept of the graph of $f$ ?

-12

-4

12

60

Show answer & explanation

Answer: 12

Answer

12

The correct answer is 12. Since the cubic function has

x

-intercepts at

(-3, 0)

,

(1, 0)

, and

(4, 0)

, it can be represented in factored form as

f(x) = a(x + 3)(x - 1)(x - 4)

for some constant

a

. Substituting the given coordinate

(2, -10)

into the function yields

-10 = a(2 + 3)(2 - 1)(2 - 4)

, which simplifies to

-10 = -10a

, meaning

a = 1

. The fully determined function is

f(x) = (x + 3)(x - 1)(x - 4)

. The

y

-intercept occurs when

x = 0

. Evaluating

f(0)

gives

(3)(-1)(-4) = 12

.

Step-by-Step Solution

1

Write the general factored form of the cubic function using its intercepts.

f(x) = a(x + 3)(x - 1)(x - 4)

, where

a

is a constant.

Since the graph of

f

has

x

-intercepts at

(-3, 0)

,

(1, 0)

, and

(4, 0)

, the values

-3

,

1

, and

4

are roots of the polynomial. Therefore,

(x + 3)

,

(x - 1)

, and

(x - 4)

are factors of the polynomial.

2

Substitute the point

(2, -10)

into the equation to solve for the constant coefficient

a

.

a = 1

Plugging in the given values yields

-10 = a(2 + 3)(2 - 1)(2 - 4)

, which simplifies to

-10 = a(5)(1)(-2)

, or

-10 = -10a

. Dividing both sides by

-10

gives

a = 1

.

3

Calculate the

y

-intercept of the function by evaluating it at

x = 0

.

f(0) = 12

The

y

-intercept occurs where the graph crosses the

y

-axis, which corresponds to

x = 0

. Substituting

x = 0

into the function

f(x) = (x + 3)(x - 1)(x - 4)

gives

f(0) = (3)(-1)(-4) = 12

.

Key Concept

Polynomial Factors and Graphs

Estimated Time:1m 30s

Question 395Question →

The quadratic equation $x^2 + 10x + c = 7$ , where $c$ is a constant, has exactly one real solution. What is the value of $c$ ?

18

25

32

-18

Show answer & explanation

Answer: $32$

Answer

32

For the quadratic equation to have exactly one real solution, it must be written in the standard form

x^2 + 10x + (c - 7) = 0

, and its discriminant must equal zero. Setting the discriminant to zero gives

10^2 - 4(1)(c - 7) = 0

. Distributing the negative four yields

100 - 4c + 28 = 0

, which simplifies to

128 - 4c = 0

. Solving for the constant gives

c = 32

. Alternatively, the left side of the equation in standard form must be a perfect square trinomial, meaning

x^2 + 10x + 25 = 0

. Thus,

c - 7 = 25

, which yields

c = 32

.

Step-by-Step Solution

1

Subtract

7

from both sides of the equation to write it in the standard quadratic form

ax^2 + bx + c = 0

.

x^2 + 10x + (c - 7) = 0

To analyze the solutions of a quadratic equation using the discriminant, the equation must be in standard form.

2

Identify the coefficients and set the discriminant

b^2 - 4ac

to

0

since the equation has exactly one real solution.

10^2 - 4(1)(c - 7) = 0

A quadratic equation has exactly one real solution if and only if its discriminant is equal to zero.

3

Solve the equation

100 - 4(c - 7) = 0

for

c

by distributing the

-4

and combining like terms.

100 - 4c + 28 = 0 \implies 128 - 4c = 0 \implies 4c = 128 \implies c = 32

Isolating the variable

c

yields the value that makes the equation have exactly one real solution.

Key Concept

Discriminant of a quadratic equation

Alternative Method

Alternatively, you can complete the square. For the quadratic expression

x^2 + 10x + (c - 7)

to have exactly one real solution, it must be a perfect square trinomial of the form

(x + d)^2 = x^2 + 2dx + d^2

. Comparing coefficients,

2d = 10 \implies d = 5

, so the constant term must be

d^2 = 25

. Setting the constant term

c - 7 = 25

yields

c = 32

.

Estimated Time:1m 30s

Question 396Question →

The solutions to the quadratic equation $x^2 - 6x - 11 = 0$ can be written in the form $x = a \pm \sqrt{b}$ , where $a$ and $b$ are integers. What is the value of $a + b$ ?

8

17

23

43

Show answer & explanation

Answer: 23

Answer

23

Completing the square on the quadratic equation

x^2 - 6x - 11 = 0

yields the solutions

x = 3 \pm \sqrt{20}

. Comparing this to the given form

x = a \pm \sqrt{b}

identifies

a = 3

and

b = 20

. Summing these values gives

a + b = 23

.

Step-by-Step Solution

1

Add 11 to both sides of the equation to isolate the variable terms.

x^2 - 6x = 11

Preparing the quadratic equation to complete the square by separating constant terms.

2

Complete the square by adding the square of half the coefficient of

x

to both sides.

x^2 - 6x + 9 = 11 + 9

, which simplifies to

(x - 3)^2 = 20

Adding

(\frac{-6}{2})^2 = 9

creates a perfect square trinomial on the left side.

3

Take the square root of both sides and solve for

x

.

x - 3 = \pm\sqrt{20}

, which gives

x = 3 \pm \sqrt{20}

Taking the square root cancels the exponent and yields two possible solutions.

4

Compare the solutions to the given form

x = a \pm \sqrt{b}

to determine

a

and

b

, and calculate their sum.

a = 3

,

b = 20

, so

a + b = 3 + 20 = 23

Matching the template shows that

a

corresponds to

3

and

b

corresponds to

20

.

Key Concept

Solving quadratic equations by completing the square and matching solutions to a given form.

Alternative Method

Instead of completing the square, you can use the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

with

a = 1

,

b = -6

, and

c = -11

. This yields

x = \frac{6 \pm \sqrt{36 - 4(1)(-11)}}{2} = \frac{6 \pm \sqrt{80}}{2} = 3 \pm \frac{\sqrt{80}}{2} = 3 \pm \sqrt{20}

. Comparing this to the given form yields

a = 3

and

b = 20

, which sums to

23

.

Estimated Time:1m 30s

Question 397Question →

In the quadratic equation $2x^2 - 15x + c = 0$ , $c$ is a constant. If one of the solutions to the equation is $x = 6$ , what is the other solution?

Show answer & explanation

Answer: 1.5

Answer

1.5

The sum of the roots of a quadratic equation in the form

ax^2 + bx + c = 0

is given by

-\frac{b}{a}

. For the equation

2x^2 - 15x + c = 0

, the sum of the roots is

-\frac{-15}{2} = 7.5

. Since one of the solutions is

6

, the other solution is

7.5 - 6 = 1.5

. Alternatively, substituting

x = 6

into the equation yields

2(6)^2 - 15(6) + c = 0 \implies 72 - 90 + c = 0 \implies c = 18

. The equation becomes

2x^2 - 15x + 18 = 0

, which factors as

(2x - 3)(x - 6) = 0

. Setting the factor

2x - 3 = 0

gives the other solution,

x = 1.5

(or

\frac{3}{2}

).

Step-by-Step Solution

1

Find the sum of the roots of the quadratic equation using Vieta's formulas.

The sum of the roots is

-\frac{-15}{2} = 7.5

.

For any quadratic equation

ax^2 + bx + c = 0

, the sum of its roots is equal to

-\frac{b}{a}

.

2

Set up an equation using the known root to find the unknown root.

6 + x_2 = 7.5

, where

x_2

is the other root.

Since one root is

6

, adding it to the second root must equal the sum of the roots.

3

Solve for the second root.

x_2 = 1.5

.

Subtracting

6

from both sides of the equation yields

1.5

.

Key Concept

Sum of roots of a quadratic equation (Vieta's Formulas)

Estimated Time:1m 30s

Question 398Question →

The function $f$ is defined by $f(x) = -x^2 + 6x - 1$ . The function $g$ is defined by $g(x) = f(x + 2) - 5$ . If the maximum value of $g(x)$ in the $xy$ -plane occurs at the point $(h, k)$ , what is the value of $h + k$ ?

Show answer & explanation

Answer: 4

Answer

4

The vertex of the original quadratic function

f(x) = -x^2 + 6x - 1

is located at

(3, 8)

. The transformation

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

translates the graph horizontally to the left by

2

units and vertically down by

5

units. This moves the vertex from

(3, 8)

to

(3 - 2, 8 - 5) = (1, 3)

. Therefore,

h = 1

and

k = 3

, and their sum

h + k

equals

4

.

Step-by-Step Solution

1

Find the vertex of the function

f(x) = -x^2 + 6x - 1

.

The vertex of

f(x)

is at

(3, 8)

.

By writing

f(x)

in vertex form,

f(x) = -(x - 3)^2 + 8

, we find that the maximum value of

f(x)

occurs at

(3, 8)

.

2

Determine the vertex

(h, k)

of the transformed function

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

.

(h, k) = (1, 3)

The horizontal shift of

f(x + 2)

translates the graph to the left by

2

units, changing the

x

-coordinate from

3

to

3 - 2 = 1

. The vertical shift of

- 5

translates the graph down by

5

units, changing the

y

-coordinate from

8

to

8 - 5 = 3

.

3

Calculate the value of

h + k

.

4

Adding the coordinates of the transformed vertex yields

h + k = 1 + 3 = 4

.

Key Concept

Vertex form of a quadratic function and translation of functions.

Question 399Question →

In the quadratic equation $3x^2 - 18x + c = 0$ , $c$ is a constant. If the sum of the squares of the two real solutions to this equation is $26$ , what is the value of $c$ ?

Show answer & explanation

Answer: 15

Answer

The value of

c

is

15

.

To find the value of

c

, we apply Vieta's formulas to the equation

3x^2 - 18x + c = 0

. The sum of the solutions is

x_1 + x_2 = -\frac{-18}{3} = 6

, and the product of the solutions is

x_1 x_2 = \frac{c}{3}

. Using the identity

x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2

, we substitute the given values:

26 = 6^2 - 2\left(\frac{c}{3}\right)

. This simplifies to

26 = 36 - \frac{2c}{3}

. Subtracting

36

from both sides yields

-10 = -\frac{2c}{3}

, which gives

2c = 30

, and thus

c = 15

. Alternatively, since the vertex of the corresponding parabola is at

x = -\frac{-18}{2(3)} = 3

, the two solutions can be represented symmetrically as

3 + d

and

3 - d

. The sum of their squares is

(3 + d)^2 + (3 - d)^2 = (9 + 6d + d^2) + (9 - 6d + d^2) = 18 + 2d^2

. Setting this equal to the given value of

26

yields

18 + 2d^2 = 26 \implies 2d^2 = 8 \implies d^2 = 4 \implies d = 2

. Therefore, the solutions are

3 + 2 = 5

and

3 - 2 = 1

. Substituting either solution back into the original equation, such as

x = 1

, yields

3(1)^2 - 18(1) + c = 0 \implies -15 + c = 0 \implies c = 15

.

Step-by-Step Solution

1

Find the sum and product of the solutions to the quadratic equation.

The sum of the solutions is

x_1 + x_2 = 6

, and their product is

x_1 x_2 = \frac{c}{3}

.

According to Vieta's formulas, for a quadratic equation

ax^2 + bx + c = 0

, the sum of the solutions is

-\frac{b}{a}

and the product of the solutions is

\frac{c}{a}

.

2

Relate the sum of the squares of the solutions to their sum and product using the algebraic identity.

x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2 \implies 26 = (6)^2 - 2\left(\frac{c}{3}\right)

.

The algebraic identity

(x_1 + x_2)^2 = x_1^2 + 2x_1 x_2 + x_2^2

can be rearranged to express the sum of squares

x_1^2 + x_2^2

in terms of the sum and product.

3

Solve the resulting linear equation for the constant

c

.

26 = 36 - \frac{2c}{3} \implies \frac{2c}{3} = 10 \implies c = 15

.

Subtracting

36

from both sides gives

-10 = -\frac{2c}{3}

. Multiplying by

-3

gives

30 = 2c

, which yields

c = 15

.

Key Concept

Vieta's Formulas and Algebraic Identities

Question 400Question →

In the $xy$ -plane, the graph of the quadratic function $f$ is a parabola with vertex $(3, -8)$ and passes through the point $(1, 0)$ . If the graph intersects the $y$ -axis at $(0, c)$ , what is the value of $c$ ?

Show answer & explanation

Answer: 10

Answer

The correct answer is 10.

By writing the function in vertex form

f(x) = a(x - h)^2 + k

and substituting the vertex

(3, -8)

, we obtain

f(x) = a(x - 3)^2 - 8

. Substituting the point

(1, 0)

into the equation gives

0 = a(1 - 3)^2 - 8

, which simplifies to

4a = 8

and yields

a = 2

. With the function fully defined as

f(x) = 2(x - 3)^2 - 8

, the

y

-intercept is found by calculating

f(0) = 2(0 - 3)^2 - 8 = 2(9) - 8 = 10

. Thus,

c = 10

.

Step-by-Step Solution

1

Express the quadratic function in vertex form using the given vertex

(3, -8)

.

f(x) = a(x - 3)^2 - 8

The vertex form of a quadratic function is

f(x) = a(x - h)^2 + k

, where

(h, k)

is the vertex.

2

Substitute the coordinates of the point

(1, 0)

into the function to solve for the leading coefficient

a

.

a = 2

Since the graph passes through

(1, 0)

, substituting

x = 1

and

f(x) = 0

must satisfy the equation.

3

Substitute

x = 0

into the completed function

f(x) = 2(x - 3)^2 - 8

to find the value of

c

.

c = 10

The intersection with the

y

-axis occurs where the input

x

is 0.

Key Concept

Writing and evaluating quadratic functions using the vertex form

Advanced Math

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

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

Alternative Method

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