Advanced Math Practice Questions — SAT — Page 22

The table below shows several values of the function $f$ .

$x$	$f(x)$
$-5$	$12$
$-3$	$-2$
$-1$	$4$
$3$	$18$

The function $g$ is defined by $g(x) = \frac{1}{2}f(x + 4) + 3$ . What is the value of $g(-5)$ ?

Show answer & explanation

Answer: 5

Answer

The correct answer is 5.

To find the value of

g(-5)

, substitute

-5

for

x

in the definition of

g(x)

, which gives

g(-5) = \frac{1}{2}f(-5 + 4) + 3 = \frac{1}{2}f(-1) + 3

. According to the table,

f(-1) = 4

. Substituting this value into the expression yields

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

.

Step-by-Step Solution

1

Substitute the input value into the function definition.

g(-5) = \frac{1}{2}f(-5 + 4) + 3 = \frac{1}{2}f(-1) + 3

To evaluate

g(-5)

, replace all occurrences of

x

with

-5

in the definition of

g(x)

.

2

Find the value of

f(-1)

from the table.

f(-1) = 4

The table provides specific input-output pairs for the function

f

. When the input is

-1

, the output is

4

.

3

Calculate the final value.

g(-5) = 2 + 3 = 5

Substitute

4

for

f(-1)

in the expression and simplify the terms.

Key Concept

Evaluating transformed functions using tabular data and applying horizontal translations, vertical compressions, and vertical translations.

Question 422Question →

The graph of the polynomial function $f$ in the $xy$ -plane is tangent to the $x$ -axis at $(1, 0)$ and crosses the $x$ -axis only at $(-2, 0)$ and $(4, 0)$ . If the $y$ -intercept of the graph of $f$ is $(0, 8)$ , what is the value of $f(-1)$ ?

-20

10

20

60

Show answer & explanation

Answer: 20

Answer

20

The graph of the polynomial function is tangent to the

x

-axis at

(1, 0)

and crosses it at

(-2, 0)

and

(4, 0)

. This means

x = 1

is a root with multiplicity 2, while

x = -2

and

x = 4

are roots with multiplicity 1. Thus, we can write the function in the form

f(x) = a(x + 2)(x - 1)^2(x - 4)

. Substituting the

y

-intercept

(0, 8)

gives

8 = a(0 + 2)(0 - 1)^2(0 - 4) \implies 8 = -8a \implies a = -1

. Substituting

a = -1

back into the formula yields

f(x) = -(x + 2)(x - 1)^2(x - 4)

. Evaluating the function at

x = -1

gives

f(-1) = -(-1 + 2)(-1 - 1)^2(-1 - 4) = -(1)(4)(-5) = 20

. Therefore, the correct value is 20.

Step-by-Step Solution

1

Determine the factored form of the polynomial from its x-intercepts and multiplicities.

f(x) = a(x + 2)(x - 1)^2(x - 4)

Since the graph is tangent to the

x

-axis at

(1, 0)

, the root

x = 1

has a multiplicity of 2, corresponding to the factor

(x - 1)^2

. Since it crosses the

x

-axis at

(-2, 0)

and

(4, 0)

, these roots have a multiplicity of 1, corresponding to the factors

(x + 2)

and

(x - 4)

.

2

Use the y-intercept to find the constant coefficient

a

.

a = -1

The

y

-intercept is

(0, 8)

, so substituting

x = 0

into the equation gives

8 = a(0 + 2)(0 - 1)^2(0 - 4) \implies 8 = -8a \implies a = -1

.

3

Substitute the value of

a

back into the function and evaluate

f(-1)

.

f(-1) = 20

Substituting

a = -1

gives

f(x) = -(x + 2)(x - 1)^2(x - 4)

. Evaluating at

x = -1

yields

f(-1) = -(-1 + 2)(-1 - 1)^2(-1 - 4) = -(1)(-2)^2(-5) = -(1)(4)(-5) = 20

.

Key Concept

Determining a polynomial function's equation from its graphical features (intercepts and tangencies) and evaluating it at a point.

Estimated Time:1m 30s

Question 423Question →

The graph of the quadratic function $f$ in the $xy$ -plane is defined by $f(x) = x^2 - 6x + 5$ . If the graph of $f$ is translated $3$ units to the left and $2$ units down, which of the following equations defines the resulting function $g$ ?

g(x) = x^2 - 12x + 30

g(x) = x^2 - 6

g(x) = x^2 - 2

g(x) = x^2 - 12x + 34

Show answer & explanation

Answer: $g(x) = x^2 - 6$

Answer

The equation that defines the function

g

is

g(x) = x^2 - 6

.

To find the function

g(x)

, we apply the transformations to

f(x) = x^2 - 6x + 5

. A translation of

3

units to the left replaces

x

with

x + 3

, which gives

f(x + 3) = (x + 3)^2 - 6(x + 3) + 5 = x^2 + 6x + 9 - 6x - 18 + 5 = x^2 - 4

. Next, translating the graph

2

units down corresponds to subtracting

2

from the function, resulting in

g(x) = (x^2 - 4) - 2 = x^2 - 6

.

Step-by-Step Solution

1

Rewrite the function

f(x)

in vertex form by completing the square.

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

Converting to vertex form

a(x - h)^2 + k

makes it straightforward to apply horizontal and vertical shifts based on the vertex

(h, k)

.

2

Apply the horizontal translation of

3

units to the left by replacing

x

with

x + 3

in the function.

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

A horizontal shift of

c

units to the left is represented by replacing

x

with

x + c

.

3

Apply the vertical translation of

2

units down by subtracting

2

from the expression obtained in the previous step.

g(x) = (x^2 - 4) - 2 = x^2 - 6

A vertical shift of

d

units downward is represented by subtracting

d

from the function's output.

Key Concept

Translations of quadratic functions in the coordinate plane.

Estimated Time:1m 30s

Question 424Question →

In the $xy$ -plane, the vertex of the parabola defined by $y = a(x - 2)(x - 8)$ has a $y$ -coordinate of $-18$ , where $a$ is a positive constant. What is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

2

The correct answer is 2. The quadratic function is given in factored form as

y = a(x - 2)(x - 8)

. The x-intercepts of this parabola are at

x = 2

and

x = 8

. Because of the symmetry of a parabola, the x-coordinate of the vertex is the midpoint of the x-intercepts:

x = \frac{2 + 8}{2} = 5

. The y-coordinate of the vertex is given as

-18

, meaning the vertex is at the point

(5, -18)

. Substituting these coordinates into the equation gives

-18 = a(5 - 2)(5 - 8)

, which simplifies to

-18 = a(3)(-3) = -9a

. Solving for

a

yields

a = 2

.

Step-by-Step Solution

1

Identify the x-intercepts from the factored form equation

y = a(x - 2)(x - 8)

and find the x-coordinate of the vertex.

The x-intercepts are

x = 2

and

x = 8

. The x-coordinate of the vertex is the midpoint of the intercepts:

x = \frac{2 + 8}{2} = 5

.

The axis of symmetry of a parabola passes through its vertex and lies midway between its x-intercepts.

2

Substitute the coordinates of the vertex

(5, -18)

into the quadratic equation to solve for the constant

a

.

Substituting

x = 5

and

y = -18

yields

-18 = a(5 - 2)(5 - 8)

, which simplifies to

-18 = a(3)(-3)

, so

-18 = -9a

, giving

a = 2

.

Since the vertex is a point on the parabola, its coordinates must satisfy the equation of the parabola.

Key Concept

Using the symmetry of quadratic functions in factored form to find the vertex coordinates.

Question 425Question →

In the $xy$ -plane, the graph of the polynomial function $f$ has $x$ -intercepts at $(-3, 0)$ , $(1, 0)$ , and $(k, 0)$ , where $k$ is a constant greater than $1$ . The function is defined by $f(x) = -2(x + 3)(x - 1)(x - k)$ . If the graph of $f$ passes through the point $(2, 20)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is

4

.

Substituting the coordinates of the point

(2, 20)

into the function equation

f(x) = -2(x + 3)(x - 1)(x - k)

gives

20 = -2(2 + 3)(2 - 1)(2 - k)

. Simplifying this expression yields

20 = -10(2 - k)

. Dividing both sides by

-10

gives

-2 = 2 - k

, which simplifies to

k = 4

. Since

4 > 1

, this meets the condition that

k

is a constant greater than

1

.

Step-by-Step Solution

1

Substitute the coordinates of the point

(2, 20)

into the function definition.

20 = -2(2 + 3)(2 - 1)(2 - k)

Since the graph passes through the point

(2, 20)

, the coordinates satisfy the equation of the function.

2

Simplify the numerical factors on the right side of the equation.

20 = -10(2 - k)

Calculating

2+3 = 5

and

2-1 = 1

, then multiplying the constants:

-2 \times 5 \times 1 = -10

.

3

Solve the linear equation for

k

.

k = 4

Dividing both sides by

-10

yields

2 - k = -2

. Adding

k

to both sides and adding

2

to both sides gives

k = 4

.

Key Concept

Determining a constant root of a polynomial function by evaluating it at a given point on its graph.

Estimated Time:1m 30s

Question 426Question →

The graph of the quadratic function $f$ in the $xy$ -plane has its vertex at $(3, 2)$ . If the graph passes through the point $(5, -6)$ , what is the $y$ -intercept of the graph of $f$ ?

-16

-11

-1

2

Show answer & explanation

Answer: $-16$

Answer

The y-intercept of the graph of

f

is

-16

.

The correct answer is found by setting up the vertex form of the quadratic function,

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

, using the vertex

(3, 2)

. Substituting the point

(5, -6)

yields

-6 = a(2)^2 + 2

, which simplifies to

-8 = 4a

, giving

a = -2

. The complete equation is

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

. Evaluating this function at

x = 0

yields

-2(0-3)^2 + 2 = -18 + 2 = -16

.

Step-by-Step Solution

1

Write the quadratic function in vertex form.

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

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex. Here,

(h, k) = (3, 2)

.

2

Substitute the coordinates of the given point

(5, -6)

to solve for the coefficient

a

.

a = -2

Plugging

x=5

and

f(x)=-6

into the equation gives

-6 = a(5-3)^2 + 2

, which simplifies to

-6 = 4a + 2

. Subtracting

2

from both sides gives

-8 = 4a

, so

a = -2

.

3

Substitute

a = -2

back into the vertex form to write the full equation of the function.

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

This represents the specific quadratic function described.

4

Evaluate the function at

x = 0

to find the y-intercept.

f(0) = -16

The y-intercept of a graph is the point where

x=0

. Evaluating

f(0)

gives

-2(0-3)^2 + 2 = -2(9) + 2 = -16

.

Key Concept

Writing a quadratic function in vertex form

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

and finding the y-intercept by evaluating the function at

x=0

.

Estimated Time:1m 30s

Question 427Question →

A polynomial $f(x)$ is defined by $f(x) = 2x^3 + hx^2 - 7x - 6$ , where $h$ is a constant. If the graph of $y = f(x)$ in the $xy$ -plane passes through the point $(2, 0)$ , what is the value of $h$ ?

Show answer & explanation

Answer: 1

Answer

The value of the constant

h

is 1.

Since the graph of

y = f(x)

passes through the point

(2, 0)

, the value of the function at

x = 2

must be

0

. Substituting

x = 2

into the equation yields

2(2)^3 + h(2)^2 - 7(2) - 6 = 0

. Simplifying the terms gives

2(8) + 4h - 14 - 6 = 0

, which simplifies to

16 + 4h - 20 = 0

, or

4h - 4 = 0

. Solving for

h

yields

h = 1

.

Step-by-Step Solution

1

Apply the point condition to the polynomial function.

f(2) = 0

Since the point

(2, 0)

lies on the graph of

y = f(x)

, substituting

x = 2

must yield

y = 0

.

2

Substitute

x = 2

into the expression for

f(x)

.

f(2) = 2(2)^3 + h(2)^2 - 7(2) - 6

We replace each occurrence of

x

with

2

to evaluate the function value.

3

Simplify the algebraic expression.

16 + 4h - 20 = 0 \implies 4h - 4 = 0

Evaluating the exponents and multiplications gives

2(8) + 4h - 14 - 6 = 16 + 4h - 20 = 4h - 4

.

4

Solve the linear equation for

h

.

h = 1

Adding 4 to both sides gives

4h = 4

, and dividing by 4 gives

h = 1

.

Key Concept

Connecting graphical x-intercepts of a polynomial to its algebraic roots and evaluation

Question 428Question →

For a cubic polynomial function $p$ , the graph of $y = p(x)$ in the $xy$ -plane has $x$ -intercepts at $(1, 0)$ and $(4, 0)$ , and $x + 3$ is a factor of $p(x)$ . If $p(0) = -24$ , what is the value of $p(2)$ ?

-10

4

20

-20

Show answer & explanation

Answer: 20

Answer

20

The correct answer is 20. By translating the factor

x + 3

and the intercepts

(1, 0)

and

(4, 0)

into roots, the polynomial can be modeled as

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

. Solving for the leading coefficient using

p(0) = -24

yields

a = -2

. Substituting

x = 2

into the resulting equation

p(x) = -2(x + 3)(x - 1)(x - 4)

gives

p(2) = -2(5)(1)(-2) = 20

.

Step-by-Step Solution

1

Identify the roots of the polynomial from the factors and intercepts.

The roots of

p(x)

are

x = -3

,

x = 1

, and

x = 4

.

A factor of

x + 3

corresponds to a root of

-3

. The

x

-intercepts at

(1, 0)

and

(4, 0)

correspond to roots at

1

and

4

.

2

Write the general factored form of the cubic polynomial.

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

A cubic polynomial with known roots

r_1, r_2, r_3

can be expressed as

p(x) = a(x-r_1)(x-r_2)(x-r_3)

.

3

Use the value

p(0) = -24

to find the constant coefficient

a

.

a = -2

Substituting

x = 0

gives

p(0) = a(0 + 3)(0 - 1)(0 - 4) = 12a

. Since

12a = -24

, we solve to find

a = -2

.

4

Evaluate

p(2)

using the fully defined polynomial.

p(2) = 20

Substituting

x = 2

into

p(x) = -2(x + 3)(x - 1)(x - 4)

yields

p(2) = -2(2 + 3)(2 - 1)(2 - 4) = -2(5)(1)(-2) = 20

.

Key Concept

Using polynomial roots, factors, and given coordinates to define and evaluate polynomial functions.

Question 429Question →

The function $h(t) = -16t^2 + v_0 t + h_0$ models the height $h(t)$ , in feet, of a model rocket $t$ seconds after launch, where $v_0$ and $h_0$ are constants. The rocket reaches its maximum height of $100$ feet above the ground $2$ seconds after it is launched. Which of the following equations defines $h(t)$ ?

h(t) = -16t^2 + 64t + 36

h(t) = -16t^2 - 64t + 36

h(t) = -16t^2 + 64t + 100

h(t) = -16t^2 + 64t - 164

Show answer & explanation

Answer: $h(t) = -16t^2 + 64t + 36$

Answer

h(t) = -16t^2 + 64t + 36

The correct equation is

h(t) = -16t^2 + 64t + 36

. Since the maximum height of

100

feet is reached at

t = 2

seconds, the vertex of the parabola is

(2, 100)

. In vertex form, a quadratic function is written as

h(t) = a(t - d)^2 + c

, where

(d, c)

is the vertex. Given that the leading coefficient

a

is

-16

, substituting the vertex yields

h(t) = -16(t - 2)^2 + 100

. Expanding this expression gives

h(t) = -16(t^2 - 4t + 4) + 100 = -16t^2 + 64t - 64 + 100 = -16t^2 + 64t + 36

.

Step-by-Step Solution

1

Identify the vertex from the problem description.

The vertex of the parabola is

(2, 100)

, representing the time

t = 2

seconds when the maximum height of

100

feet is reached.

The vertex

(h, k)

of a quadratic function represents the maximum or minimum point of its graph.

2

Write the quadratic equation in vertex form.

h(t) = a(t - 2)^2 + 100

. Since the coefficient of

t^2

in the standard form is

-16

, we set

a = -16

, giving

h(t) = -16(t - 2)^2 + 100

.

The vertex form of a quadratic function is

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

, and the leading coefficient

a

is the same as in standard form.

3

Expand the vertex form equation into standard form.

h(t) = -16(t^2 - 4t + 4) + 100 = -16t^2 + 64t - 64 + 100 = -16t^2 + 64t + 36

.

Expanding the squared term and distributing the leading coefficient converts the vertex form to standard form

y = ax^2 + bx + c

.

Key Concept

Quadratic Functions and Graphs

Alternative Method

The axis of symmetry for a quadratic function in standard form

y = ax^2 + bx + c

is given by

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

. For this model, the maximum occurs at

t = 2

, meaning the axis of symmetry is

t = 2

. Since

a = -16

, we have

2 = -\frac{b}{2(-16)}

, which simplifies to

b = 64

. We can then test the remaining options where the linear coefficient is

64

. Substituting

t = 2

into the correct equation

h(t) = -16t^2 + 64t + 36

yields

h(2) = -16(4) + 64(2) + 36 = -64 + 128 + 36 = 100

, confirming it reaches the correct maximum height.

Estimated Time:1m 30s

Question 430Question →

For the polynomial $p(x) = 2x^3 - x^2 - kx + 6$ , where $k$ is a constant, the remainder when $p(x)$ is divided by $2x - 3$ is $0$ . What is the value of $k$ ?

Show answer & explanation

Answer: 7

Answer

The value of

k

is

7

.

The correct answer is

7

. According to the Factor Theorem, a linear expression

ax - b

is a factor of a polynomial

p(x)

if and only if

p\left(\frac{b}{a}\right) = 0

. Here, the divisor is

2x - 3

, so setting

2x - 3 = 0

gives the root

x = \frac{3}{2}

. Substituting

x = \frac{3}{2}

into

p(x) = 2x^3 - x^2 - kx + 6

gives

2\left(\frac{3}{2}\right)^3 - \left(\frac{3}{2}\right)^2 - k\left(\frac{3}{2}\right) + 6 = 0

. Simplifying this equation gives

\frac{27}{4} - \frac{9}{4} - \frac{3}{2}k + 6 = 0

, which reduces to

\frac{9}{2} - \frac{3}{2}k + 6 = 0

. Combining the constant terms gives

\frac{21}{2} - \frac{3}{2}k = 0

, which simplifies to

3k = 21

. Solving for

k

gives

k = 7

.

Step-by-Step Solution

1

Determine the root associated with the linear factor

2x - 3

.

Setting

2x - 3 = 0

gives

x = \frac{3}{2}

. By the Factor Theorem,

p\left(\frac{3}{2}\right) = 0

.

According to the Factor Theorem, a polynomial

p(x)

has a factor of the form

ax - b

if and only if

p\left(\frac{b}{a}\right) = 0

.

2

Substitute

x = \frac{3}{2}

into the polynomial expression

2x^3 - x^2 - kx + 6

and set it to

0

.

2\left(\frac{3}{2}\right)^3 - \left(\frac{3}{2}\right)^2 - k\left(\frac{3}{2}\right) + 6 = 0

This establishes a linear equation in terms of the unknown constant

k

.

3

Simplify the numerical terms and solve for

k

.

2\left(\frac{27}{8}\right) - \frac{9}{4} - \frac{3}{2}k + 6 = 0 \implies \frac{27}{4} - \frac{9}{4} - \frac{3}{2}k + 6 = 0 \implies \frac{9}{2} - \frac{3}{2}k + 6 = 0 \implies \frac{21}{2} - \frac{3}{2}k = 0 \implies 3k = 21 \implies k = 7

.

Standard fractional arithmetic and algebraic isolation are used to find the value of

k

.

Key Concept

Factor and Remainder Theorems

Question 431Question →

In the $xy$ -plane, the graph of the quadratic function $f$ has a vertex at $(h, k)$ , where $h$ and $k$ are constants. The graph passes through the points $(1, 10)$ and $(7, 10)$ . If the minimum value of the function $f$ is $1$ , what is the value of $f(2)$ ?

Show answer & explanation

Answer: 5

Answer

The value of

f(2)

is 5.

The correct answer is 5. Since the graph of the quadratic function passes through

(1, 10)

and

(7, 10)

, the axis of symmetry is the vertical line halfway between

x = 1

and

x = 7

, which is

x = \frac{1+7}{2} = 4

. The minimum value of the function is

1

, which occurs at the vertex, so the vertex is

(4, 1)

. In vertex form, the function is

f(x) = a(x-4)^2 + 1

. Substituting the point

(1, 10)

yields

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

, which simplifies to

9 = 9a

, so

a = 1

. The function is

f(x) = (x-4)^2 + 1

. Evaluating this function at

x = 2

gives

f(2) = (2-4)^2 + 1 = 5

.

Step-by-Step Solution

1

Find the

x

-coordinate of the vertex using the symmetry of the parabola.

h = 4

A parabola is symmetric about its vertical axis of symmetry. Since the points

(1, 10)

and

(7, 10)

have the same

y

-coordinate, the axis of symmetry is exactly halfway between their

x

-coordinates:

x = \frac{1 + 7}{2} = 4

.

2

Find the vertex coordinates using the minimum value of the function.

Vertex is

(4, 1)

The vertex of a quadratic function with a minimum value lies on the axis of symmetry, and its

y

-coordinate is the minimum value. Thus, the vertex

(h, k)

is

(4, 1)

.

3

Write the vertex form of the quadratic function and solve for the leading coefficient

a

.

f(x) = (x - 4)^2 + 1

Using the vertex form

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

, we substitute

h = 4

and

k = 1

to get

f(x) = a(x - 4)^2 + 1

. Substituting the point

(1, 10)

gives

10 = a(1 - 4)^2 + 1 \implies 9 = 9a \implies a = 1

.

4

Evaluate

f(2)

using the determined function.

f(2) = 5

Substitute

x = 2

into the equation

f(x) = (x - 4)^2 + 1

to obtain

f(2) = (2 - 4)^2 + 1 = 4 + 1 = 5

.

Key Concept

Using symmetry and the vertex form of a quadratic function to determine its equation and evaluate values.

Estimated Time:1m 30s

Question 432Question →

The quadratic function $f$ is defined by $f(x) = (x - 4)(x - 10)$ . In the $xy$ -plane, the graph of function $g$ is obtained by translating the graph of $f$ horizontally so that the vertex of the graph of $g$ lies on the $y$ -axis. Which of the following equations defines the function $g$ ?

g(x) = x^2 - 14x + 40

g(x) = x^2 - 9

g(x) = x^2 - 28x + 187

g(x) = x^2 + 14x + 40

Show answer & explanation

Answer: $g(x) = x^2 - 9$

Answer

The correct equation is

g(x) = x^2 - 9

.

The x-coordinate of the vertex of the graph of

f(x) = (x - 4)(x - 10)

is the average of its x-intercepts:

\frac{4 + 10}{2} = 7

. For the vertex of the graph of

g

to lie on the y-axis, the x-coordinate of the new vertex must be 0. This requires translating the graph of

f

to the left by 7 units, which corresponds to the transformation

g(x) = f(x + 7)

. Substituting

x + 7

for

x

in

f(x)

gives

g(x) = ((x + 7) - 4)((x + 7) - 10) = (x + 3)(x - 3) = x^2 - 9

.

Step-by-Step Solution

1

Identify the x-intercepts of the function

f(x) = (x - 4)(x - 10)

.

The x-intercepts are

x = 4

and

x = 10

.

The x-intercepts occur where the function equals zero, which are the roots of the factors.

2

Find the x-coordinate of the vertex of the graph of

f

.

The x-coordinate of the vertex is

x = 7

.

The axis of symmetry (and thus the x-coordinate of the vertex) is the average of the x-intercepts:

\frac{4 + 10}{2} = 7

.

3

Determine the translation needed to place the vertex on the y-axis.

Translate the graph to the left by 7 units.

The y-axis corresponds to

x = 0

. To move the vertex from

x = 7

to

x = 0

, the graph must be shifted 7 units to the left.

4

Apply the horizontal translation to find the equation for

g(x)

.

g(x) = (x + 3)(x - 3) = x^2 - 9

.

A horizontal translation of 7 units to the left is represented by

g(x) = f(x + 7)

. Substituting

x + 7

for

x

in the equation for

f(x)

gives

((x + 7) - 4)((x + 7) - 10) = (x + 3)(x - 3) = x^2 - 9

.

Key Concept

Identifying the vertex of a quadratic function from its factored form and applying horizontal translation rules.

Question 433Question →

In the $xy$ -plane, the graph of the polynomial function $g$ , defined by $g(x) = 3(x - k)(x + 2)^2$ , has a $y$ -intercept at $(0, -24)$ , where $k$ is a constant. What is the value of $k$ ?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

Evaluating the function at

x = 0

yields

g(0) = 3(-k)(2)^2 = -12k

. Setting this equal to the given

y

-intercept value of

-24

gives

-12k = -24

. Solving for

k

gives

k = 2

.

Step-by-Step Solution

1

Determine the value of the function at the

y

-intercept

g(0) = -24

The

y

-intercept of a graph in the

xy

-plane is the point where

x = 0

. Since the

y

-intercept is

(0, -24)

, the function value when

x = 0

must be

-24

.

2

Substitute

x = 0

into the function definition

g(0) = 3(0 - k)(0 + 2)^2

To evaluate the expression at

x = 0

, we substitute

0

for every instance of

x

in the equation.

3

Simplify the algebraic expression

g(0) = -12k

Simplifying the terms:

(0+2)^2 = 4

, and

3(0-k) = -3k

. Multiplying these gives

-3k \times 4 = -12k

.

4

Solve for the constant

k

k = 2

Equating the simplified expression to the known

y

-value at the intercept gives

-12k = -24

. Dividing both sides by

-12

yields

k = 2

.

Key Concept

Using the y-intercept of a polynomial function to solve for an unknown constant coefficient.

Question 434Question →

In the $xy$ -plane, the graph of $y = x^2 - 8x + 12$ represents the quadratic function $f$ . If this graph is shifted $3$ units to the left and $5$ units up to create the graph of a new function $g$ , what is the $y$ -value of the vertex of the graph of $g$ ?

Show answer & explanation

Answer: 1

Answer

The y-value of the vertex of the graph of g is 1.

The vertex of the original quadratic function f(x) = x^2 - 8x + 12 can be found by rewriting it in vertex form, which is f(x) = (x - 4)^2 - 4. This shows that the vertex of the original graph is (4, -4). A translation of 3 units to the left subtracts 3 from the x-coordinate of the vertex (4 - 3 = 1), and a translation of 5 units up adds 5 to the y-coordinate of the vertex (-4 + 5 = 1). Thus, the vertex of the graph of the new function g is (1, 1), making its y-value 1.

Step-by-Step Solution

1

Find the vertex of the original quadratic function.

The vertex of the original graph is

(4, -4)

.

By completing the square on

y = x^2 - 8x + 12

, we get

y = (x - 4)^2 - 4

, which reveals the vertex is at

(4, -4)

.

2

Apply the translation to the vertex coordinates.

The translated vertex coordinates are

(1, 1)

.

Shifting a point

(x, y)

by

3

units left and

5

units up results in the point

(x - 3, y + 5)

. Applying this to the vertex

(4, -4)

yields

(4 - 3, -4 + 5) = (1, 1)

.

3

Identify the y-coordinate of the new vertex.

The y-value is

1

.

The vertex of the graph of

g

is

(1, 1)

, where the second coordinate represents the y-value.

Key Concept

Identifying the vertex of a quadratic function and applying horizontal and vertical translations in the coordinate plane.

Question 435Question →

The polynomial $p(x)$ is defined by $p(x) = x^3 - 2x^2 - kx + 6$ , where $k$ is a constant. If $x - 3$ is a factor of $p(x)$ , which of the following is a factor of $p(x) - 2x + 8$ ?

x - 1

x - 2

x + 2

x + 3

Show answer & explanation

Answer: $x - 2$

Answer

The expression

x - 2

is a factor of the modified polynomial.

The correct answer is the binomial expression

x - 2

. By the Factor Theorem,

x - 3

being a factor of

p(x)

implies

p(3) = 0

. Substituting

x = 3

into

p(x) = x^3 - 2x^2 - kx + 6

gives

27 - 18 - 3k + 6 = 0

, which simplifies to

15 - 3k = 0

, yielding

k = 5

. Thus,

p(x) = x^3 - 2x^2 - 5x + 6

. The modified expression is

p(x) - 2x + 8 = x^3 - 2x^2 - 7x + 14

. Factoring this by grouping gives

x^2(x - 2) - 7(x - 2) = (x^2 - 7)(x - 2)

, showing that

x - 2

is a factor.

Step-by-Step Solution

1

Apply the Factor Theorem to find the value of

k

. Since

x - 3

is a factor of

p(x)

, we must have

p(3) = 0

.

Substituting

x = 3

into the equation gives

3^3 - 2(3)^2 - k(3) + 6 = 0

, which simplifies to

27 - 18 - 3k + 6 = 0

, or

15 - 3k = 0

. Solving for

k

yields

k = 5

.

This allows us to determine the complete polynomial expression for

p(x)

.

2

Substitute the value of

k

back into

p(x)

and determine the expression for the modified polynomial,

q(x) = p(x) - 2x + 8

.

Substituting

k = 5

gives

p(x) = x^3 - 2x^2 - 5x + 6

. Therefore, the modified polynomial is

q(x) = (x^3 - 2x^2 - 5x + 6) - 2x + 8 = x^3 - 2x^2 - 7x + 14

.

This simplifies the expression so we can find its factors.

3

Factor the modified polynomial

q(x) = x^3 - 2x^2 - 7x + 14

by grouping.

Grouping the terms gives

x^2(x - 2) - 7(x - 2)

. Factoring out the common binomial term

(x - 2)

results in

(x^2 - 7)(x - 2)

.

This reveals the individual linear and quadratic factors of the polynomial.

4

Identify which of the options is a factor of the modified polynomial.

The factors of the polynomial are

x - 2

and

x^2 - 7

. Thus,

x - 2

is a factor.

This answers the question by matching our factored form with the options.

Key Concept

The Factor Theorem states that a linear expression

x - c

is a factor of a polynomial

f(x)

if and only if

f(c) = 0

. We can also factor cubic polynomials by grouping terms to find their roots and factors.

Alternative Method

Alternatively, instead of fully factoring the polynomial, one can apply the Factor Theorem directly to the choices. Since a linear expression

x - c

is a factor of

q(x) = p(x) - 2x + 8

if and only if

q(c) = 0

, we can substitute the root

c

corresponding to each choice into

q(x)

after finding

k = 5

. For the correct factor

x - 2

, substituting

x = 2

yields

q(2) = 2^3 - 2(2)^2 - 7(2) + 14 = 8 - 8 - 14 + 14 = 0

.

Estimated Time:1m 30s

Question 436Question →

The table below shows selected values of $x$ and the corresponding values of the polynomial function $p(x)$ .

$x$	$p(x)$
$0$	$-16$
$2$	$0$
$3$	$-4$
$4$	$0$

If $p(x) = a(x-2)(x-4)(x-k)$ for all real numbers $x$ , where $a$ and $k$ are constants, what is the value of $k$ ?

Show answer & explanation

Answer: 1

Answer

The value of

k

is

1

.

Substituting the coordinate pair

(0, -16)

into the equation

p(x) = a(x-2)(x-4)(x-k)

gives

p(0) = a(-2)(-4)(-k) = -8ak = -16

, which simplifies to

ak = 2

. Substituting the coordinate pair

(3, -4)

gives

p(3) = a(3-2)(3-4)(3-k) = -a(3-k) = -4

. Expanding this equation yields

-3a + ak = -4

. Replacing

ak

with

2

in this equation results in

-3a + 2 = -4

, which simplifies to

-3a = -6

, so

a = 2

. Using

a = 2

in the relation

ak = 2

gives

2k = 2

, which yields

k = 1

.

Step-by-Step Solution

1

Substitute the coordinates

(0, -16)

from the table into the given equation

p(x) = a(x-2)(x-4)(x-k)

to form a relationship between the constants.

p(0) = a(0-2)(0-4)(0-k) = a(-2)(-4)(-k) = -8ak = -16

, which simplifies to

ak = 2

.

To establish a relation between constants

a

and

k

using the y-intercept of the polynomial.

2

Substitute another known point from the table,

(3, -4)

, into the polynomial equation.

p(3) = a(3-2)(3-4)(3-k) = a(1)(-1)(3-k) = -a(3-k) = -4

.

To obtain a second equation relating the constants

a

and

k

.

3

Solve the system of equations by distributing

-a

in the second equation and substituting the value of

ak

from the first step.

-3a + ak = -4

. Since

ak = 2

, this becomes

-3a + 2 = -4 \implies -3a = -6 \implies a = 2

.

To solve for the leading coefficient constant

a

.

4

Substitute the value of

a

back into the first relation to solve for

k

.

Since

a = 2

and

ak = 2

, we have

2k = 2 \implies k = 1

.

To find the final value of

k

.

Key Concept

Solving for polynomial constants using given coordinate points from a table

Question 437Question →

A cubic polynomial function $f$ has $x$ -intercepts at $(2, 0)$ , $(5, 0)$ , and $(c, 0)$ , where $c$ is a positive constant. In the $xy$ -plane, the graph of $y = f(x)$ has a $y$ -intercept at $(0, 60)$ . If $f(1) = 16$ , what is the value of $c$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

The factored form of a cubic polynomial with

x

-intercepts at

x = 2

,

x = 5

, and

x = c

is

f(x) = a(x-2)(x-5)(x-c)

. Evaluating the function at the

y

-intercept

x = 0

gives

f(0) = a(-2)(-5)(-c) = -10ac = 60

, which simplifies to

ac = -6

. Using the point

(1, 16)

yields

f(1) = a(1-2)(1-5)(1-c) = 4a(1-c) = 16

. Expanding this equation gives

4a - 4ac = 16

. Substituting

ac = -6

into this equation yields

4a - 4(-6) = 16

, which simplifies to

4a + 24 = 16

, resulting in

a = -2

. Finally, substituting

a = -2

into

ac = -6

yields

-2c = -6

, so

c = 3

.

Step-by-Step Solution

1

Write the polynomial in factored form using its

x

-intercepts.

f(x) = a(x-2)(x-5)(x-c)

By the factor theorem, if a polynomial has

x

-intercepts at

x = r_1, r_2, r_3

, then

(x-r_1)

,

(x-r_2)

, and

(x-r_3)

are factors of the polynomial.

2

Use the

y

-intercept

(0, 60)

to find a relation between

a

and

c

.

ac = -6

Since the

y

-intercept is at

(0, 60)

, we substitute

x = 0

into the polynomial and set the expression equal to

60

:

a(-2)(-5)(-c) = -10ac = 60

, which gives

ac = -6

.

3

Use the given point

f(1) = 16

to set up a second equation.

4a(1-c) = 16

Substitute

x = 1

and

f(1) = 16

into the factored form:

a(1-2)(1-5)(1-c) = a(-1)(-4)(1-c) = 4a(1-c) = 16

.

4

Solve for the leading coefficient

a

by substituting

ac = -6

.

a = -2

Expanding the equation from Step 3 yields

4a - 4ac = 16

. Substituting

ac = -6

gives

4a - 4(-6) = 16

, which simplifies to

4a + 24 = 16

, so

4a = -8

and

a = -2

.

5

Solve for the constant

c

.

c = 3

Using the relation

ac = -6

and substituting

a = -2

gives

-2c = -6

, which yields

c = 3

.

Key Concept

Using the factor theorem to set up a cubic polynomial equation and solving for unknown parameters using given points.

Question 438Question →

A polynomial $f(x)$ has a remainder of $12$ when divided by $x - 4$ . If $g(x) = (x - 2)f(x - 2) - 5$ , what is the remainder when $g(x)$ is divided by $x - 6$ ?

7

19

43

-53

Show answer & explanation

Answer: 43

Answer

The correct answer is 43, which is the remainder when g(x) is divided by x - 6.

According to the Remainder Theorem, since the polynomial

f(x)

has a remainder of

12

when divided by

x - 4

, we have

f(4) = 12

. To find the remainder when

g(x)

is divided by

x - 6

, we apply the Remainder Theorem again to evaluate

g(6)

. Substituting

x = 6

into the definition of

g(x)

gives

g(6) = (6 - 2)f(6 - 2) - 5 = 4f(4) - 5

. Substituting

f(4) = 12

yields

4(12) - 5 = 48 - 5 = 43

.

Step-by-Step Solution

1

Apply the Remainder Theorem to the polynomial

f(x)

.

f(4) = 12

The Remainder Theorem states that the remainder when a polynomial

f(x)

is divided by

x - c

is equal to

f(c)

.

2

Set up the expression for the remainder of

g(x)

when divided by

x - 6

.

Evaluate

g(6)

By the Remainder Theorem, the remainder of

g(x)

divided by

x - 6

is equal to

g(6)

.

3

Substitute

x = 6

into the definition of

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

.

g(6) = (6 - 2)f(6 - 2) - 5 = 4f(4) - 5

Evaluating

g(6)

requires substituting

x = 6

into all occurrences of

x

in the equation for

g(x)

.

4

Substitute the value of

f(4)

into the expression for

g(6)

and simplify.

g(6) = 4(12) - 5 = 48 - 5 = 43

Using the value of

f(4) = 12

from Step 1 allows us to calculate the numerical value of the remainder.

Key Concept

The Remainder Theorem and evaluation of composite polynomial functions.

Alternative Method

Instead of evaluating

g(6)

directly, one can write

f(x) = (x - 4)q(x) + 12

for some polynomial

q(x)

. Substituting this expression into the equation for

g(x)

gives

g(x) = (x - 2)[(x - 6)q(x - 2) + 12] - 5 = (x - 2)(x - 6)q(x - 2) + 12(x - 2) - 5

. Evaluating this expression at

x = 6

makes the first term

0

, leaving

12(6 - 2) - 5 = 12(4) - 5 = 43

.

Estimated Time:1m 30s

Advanced Math

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