Advanced Math Practice Questions — SAT — Page 16

Question 301Question →

A manufacturer models the daily profit, $P(x)$ , in dollars, from producing and selling $x$ units of a product using the function $P(x) = -2x^2 + kx - 800$ , where $k$ is a constant. If the maximum daily profit is $1,000$ dollars, what is the number of units that must be sold to achieve this maximum profit?

10

30

60

120

Show answer & explanation

Answer: 30

Answer

The manufacturer must sell 30 units to achieve the maximum daily profit.

The correct answer is 30. The maximum value of a downward-opening quadratic function occurs at its vertex

(h, q)

. Given that the maximum value is 1000, we write the function in vertex form:

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

. Expanding this gives

P(x) = -2x^2 + 4hx - 2h^2 + 1000

. Comparing this to the given function

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

, the constant term must satisfy

-2h^2 + 1000 = -800

. Solving for

h

yields

2h^2 = 1800

, so

h^2 = 900

. Taking the positive square root because the number of units must be positive gives

h = 30

.

Step-by-Step Solution

1

Write the quadratic function in vertex form and identify the given parameters.

The vertex form of a quadratic function is

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

, where

(h, q)

is the vertex. Here,

a = -2

and the maximum profit

q = 1000

, so the function is

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

.

Since the coefficient of the

x^2

term is negative (

a = -2

), the parabola opens downward, meaning the vertex represents the maximum value of the function.

2

Expand the vertex form equation to compare its coefficients with the standard form

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

.

P(x) = -2(x^2 - 2hx + h^2) + 1000 = -2x^2 + 4hx - (2h^2 - 1000)

Expanding the vertex form allows us to equate corresponding coefficients and constant terms between the two forms of the quadratic function.

3

Equate the constant terms from both expressions to solve for the vertex

x

-coordinate

h

.

-2h^2 + 1000 = -800 \Rightarrow -2h^2 = -1800 \Rightarrow h^2 = 900 \Rightarrow h = 30

(since the number of units must be positive).

By setting the constant terms equal, we can isolate and solve for

h

, which represents the number of units that maximizes the daily profit.

Key Concept

Quadratic functions in vertex form and their standard form equivalents.

Question 302Question →

In the $xy$ -plane, the vertex of the parabola with equation $y = -2x^2 + bx + c$ , where $b$ and $c$ are constants, lies on the line $y = 3x + 5$ . If the parabola has its maximum value at $x = -2$ , what is the value of $c$ ?

3

-9

-7

-1

Show answer & explanation

Answer: -9

Answer

-9

The maximum value of the quadratic function

y = -2x^2 + bx + c

occurs at its vertex. Since the maximum occurs at

x = -2

, the

x

-coordinate of the vertex is

-2

. The vertex lies on the line

y = 3x + 5

, so substituting

x = -2

into this equation gives the

y

-coordinate of the vertex:

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

. Thus, the vertex of the parabola is

(-2, -1)

. The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex. Substituting

a = -2

,

h = -2

, and

k = -1

gives

y = -2(x + 2)^2 - 1

. Expanding this equation yields

y = -2(x^2 + 4x + 4) - 1 = -2x^2 - 8x - 9

. Comparing this to the standard form

y = -2x^2 + bx + c

shows that

c = -9

.

Step-by-Step Solution

1

Identify the x-coordinate of the vertex of the parabola

The x-coordinate of the vertex is

h = -2

.

For a quadratic function in standard form with a negative leading coefficient, the maximum value occurs at the vertex. Thus, the x-coordinate of the vertex is

-2

.

2

Determine the y-coordinate of the vertex using the equation of the line

The y-coordinate of the vertex is

k = -1

.

Since the vertex

(h, k)

lies on the line

y = 3x + 5

and

h = -2

, substituting

x = -2

yields

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

.

3

Write the quadratic equation in vertex form and expand it to find c

y = -2x^2 - 8x - 9

, which gives

c = -9

.

Using the vertex form

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

with

a = -2

,

h = -2

, and

k = -1

, the equation is

y = -2(x + 2)^2 - 1

. Expanding this yields

y = -2(x^2 + 4x + 4) - 1 = -2x^2 - 8x - 9

. Comparing this with

y = -2x^2 + bx + c

shows that

c = -9

.

Key Concept

Vertex form of a quadratic function and coordinate geometry.

Estimated Time:2m 0s

Question 303Question →

For the quadratic function $f(x) = -x^2 + bx + c$ , where $b$ and $c$ are positive constants, the maximum value of $f(x)$ is $k$ . The distance between the two $x$ -intercepts of the graph of $y = f(x)$ in the $xy$ -plane is equal to $\frac{2}{3}k$ . If the graph of $y = f(x)$ passes through the point $(1, 8)$ , what is the value of $b$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is 4.

The correct answer is 4. By expressing the maximum value

k

and the distance between the

x

-intercepts

d

in terms of the constants

b

and

c

, we set up the equation

d = \frac{2}{3}k

. This simplifies to

b^2 + 4c = 36

. Using the point

(1, 8)

, we establish

c = 9 - b

. Substituting this into the first equation yields

b^2 - 4b = 0

, giving

b = 4

as the only positive solution.

Step-by-Step Solution

1

Find the maximum value

k

of the function

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

in terms of

b

and

c

.

k = \frac{b^2}{4} + c

The maximum value of a quadratic function with a negative leading coefficient occurs at its vertex, where

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

.

2

Find the distance

d

between the

x

-intercepts of the graph in terms of

b

and

c

.

d = \sqrt{b^2 + 4c}

The

x

-intercepts are the roots of

-x^2 + bx + c = 0

, which are

x = \frac{b \pm \sqrt{b^2 + 4c}}{2}

. The distance between them is the difference of these roots.

3

Use the relation

d = \frac{2}{3}k

to find the value of the expression

b^2 + 4c

.

b^2 + 4c = 36

Substituting the expressions for

d

and

k

gives

\sqrt{b^2 + 4c} = \frac{1}{6}(b^2 + 4c)

. Solving this radical equation yields

b^2 + 4c = 36

.

4

Use the point

(1, 8)

to express

c

in terms of

b

.

c = 9 - b

Since the graph passes through

(1, 8)

, substituting

x = 1

and

y = 8

into

y = -x^2 + bx + c

yields

8 = -1 + b + c

, which simplifies to

c = 9 - b

.

5

Substitute

c = 9 - b

into

b^2 + 4c = 36

and solve for

b

.

b = 4

Substituting yields

b^2 + 4(9 - b) = 36 \implies b^2 - 4b = 0

. Solving for

b

gives

b = 0

or

b = 4

. Since

b

is a positive constant, we have

b = 4

.

Key Concept

Quadratic Functions and Graphs

Question 304Question →

The polynomial function $p$ is defined by $p(x) = x^3 + bx^2 + cx + d$ , where $b$ , $c$ , and $d$ are constants. In the $xy$ -plane, the graph of $y = p(x)$ has $x$ -intercepts at $(2, 0)$ and $(-3, 0)$ . If the remainder when $p(x)$ is divided by $x - 1$ is $-8$ , what is the value of $d$ ?

-6

6

-2

18

Show answer & explanation

Answer: $-6$

Answer

-6

The correct value is

-6

. Since the graph has

x

-intercepts at

(2, 0)

and

(-3, 0)

, the Factor Theorem dictates that

(x - 2)

and

(x + 3)

are factors of

p(x)

. Since the leading coefficient of the cubic polynomial

p(x)

is

1

, it can be written in factored form as

p(x) = (x - 2)(x + 3)(x - r)

, where

r

is the third root. According to the Remainder Theorem, the remainder when

p(x)

is divided by

x - 1

is equal to

p(1) = -8

. Substituting

x = 1

yields

-4(1 - r) = -8

, which simplifies to

1 - r = 2

, so

r = -1

. The polynomial is thus

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

. The constant term

d

is equal to the value of the function when

x = 0

, which is

p(0) = (-2)(3)(1) = -6

.

Step-by-Step Solution

1

Determine the factors corresponding to the given

x

-intercepts of the polynomial.

The factors are

(x - 2)

and

(x + 3)

.

Since the graph of

y = p(x)

has

x

-intercepts at

(2, 0)

and

(-3, 0)

, we know that

p(2) = 0

and

p(-3) = 0

. According to the Factor Theorem,

(x - 2)

and

(x + 3)

must be factors of

p(x)

.

2

Set up the factored form of the cubic polynomial function

p(x)

.

p(x) = (x - 2)(x + 3)(x - r)

Because

p(x)

is a cubic polynomial (degree 3) with a leading coefficient of

1

, it can be written as the product of three linear factors:

(x - 2)

,

(x + 3)

, and

(x - r)

, where

r

is the unknown third root.

3

Apply the Remainder Theorem to solve for the third root

r

.

r = -1

By the Remainder Theorem, the remainder when

p(x)

is divided by

x - 1

is equal to

p(1)

. We are given that this remainder is

-8

, so

p(1) = -8

. Substituting

x = 1

into our factored expression gives:

p(1) = (1 - 2)(1 + 3)(1 - r) = -8 \Rightarrow -1 \cdot 4 \cdot (1 - r) = -8 \Rightarrow -4(1 - r) = -8 \Rightarrow 1 - r = 2 \Rightarrow r = -1

.

4

Determine the value of the constant coefficient

d

.

d = -6

Using the root

r = -1

, the complete factored expression is

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

. The constant term

d

is equivalent to

p(0)

:

d = p(0) = (0 - 2)(0 + 3)(0 + 1) = (-2)(3)(1) = -6

.

Key Concept

Using the Factor Theorem and Remainder Theorem to find unknown coefficients in a polynomial function.

Alternative Method

Alternatively, you can expand the general form

p(x) = (x - 2)(x + 3)(x - r) = (x^2 + x - 6)(x - r) = x^3 + (1 - r)x^2 - (r + 6)x + 6r

. Comparing this to

p(x) = x^3 + bx^2 + cx + d

, we see that

d = 6r

. Since the remainder when

p(x)

is divided by

x - 1

is

-8

, we have

p(1) = -8

. Substituting

x = 1

into our expanded form gives

1^3 + (1 - r)(1)^2 - (r + 6)(1) + 6r = -8 \Rightarrow 1 + 1 - r - r - 6 + 6r = -8 \Rightarrow 4r - 4 = -8 \Rightarrow 4r = -4 \Rightarrow r = -1

. Thus,

d = 6(-1) = -6

.

Estimated Time:2m 0s

Question 305Question →

A polynomial function $P(x)$ of degree 4 with real coefficients is symmetric about the line $x = 2$ in the $xy$ -plane. If $P(x)$ is divisible by $x^2 - 4x + 3$ , the remainder when $P(x)$ is divided by $x - 4$ is $36$ , and $P(2) = -8$ , what is the value of $P(5)$ ?

Show answer & explanation

Answer: 136

Answer

The value of

P(5)

is 136.

The correct answer of 136 is found by representing the symmetric fourth-degree polynomial as

P(x) = a(x - 2)^4 + b(x - 2)^2 + c

, solving for the coefficients using the roots at

x=3

, the point

(2, -8)

, and the remainder point

(4, 36)

, and then evaluating the resulting expression

(x-2)^4 + 7(x-2)^2 - 8

at

x=5

.

Step-by-Step Solution

1

Express the fourth-degree polynomial in a form that reflects its symmetry.

P(x) = a(x - 2)^4 + b(x - 2)^2 + c

Since the graph of

P(x)

is symmetric about the line

x = 2

, the polynomial expression in terms of

(x - 2)

must contain only even powers.

2

Use the divisibility condition to establish an equation for the coefficients.

P(3) = a(3-2)^4 + b(3-2)^2 + c = a + b + c = 0

The divisor

x^2 - 4x + 3

factors into

(x-1)(x-3)

. By the Factor Theorem,

P(3) = 0

and

P(1) = 0

.

3

Use the given value

P(2) = -8

to find the constant term

c

.

c = -8

Substituting

x = 2

into

P(x) = a(x - 2)^4 + b(x - 2)^2 + c

makes the terms with

(x-2)

equal to zero, leaving

P(2) = c

.

4

Formulate a system of linear equations for

a

and

b

.

a + b = 8

and

4a + b = 11

Substituting

c = -8

into

a + b + c = 0

gives

a + b = 8

. By the Remainder Theorem,

P(4) = 36

, which gives

a(4-2)^4 + b(4-2)^2 - 8 = 36 \implies 16a + 4b = 44 \implies 4a + b = 11

.

5

Solve the system of equations for

a

and

b

.

a = 1

and

b = 7

Subtracting

a + b = 8

from

4a + b = 11

yields

3a = 3 \implies a = 1

, which then gives

b = 7

.

6

Evaluate the polynomial at

x = 5

.

P(5) = 136

Substitute

a = 1

,

b = 7

,

c = -8

, and

x = 5

into the symmetric polynomial form:

P(5) = (5-2)^4 + 7(5-2)^2 - 8 = 3^4 + 7(3^2) - 8 = 81 + 63 - 8 = 136

.

Key Concept

Polynomial Factors and Graphs

Question 306Question →

If

x

is a real number that satisfies the equation below, what is the value of

x

?

\frac{x + 2}{x - 4} - \frac{3}{x} = \frac{12}{x^2 - 4x}

Show answer & explanation

Answer: 1

Answer

The only valid real solution to the equation is

1

.

Multiplying the equation by the common denominator

x(x - 4)

simplifies the equation to

x^2 - x = 0

. Solving this quadratic equation yields

x = 0

and

x = 1

. Because

x = 0

leads to a division by zero in the original equation, it is extraneous. Therefore, the only valid real solution is

1

.

Step-by-Step Solution

1

Determine the common denominator and restrictions for the rational equation.

The common denominator is

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

. The restrictions are

x \neq 0

and

x \neq 4

.

Finding the common denominator allows us to eliminate fractions, while the restrictions help us identify potential extraneous solutions.

2

Multiply the entire equation by the common denominator

x(x - 4)

.

x(x + 2) - 3(x - 4) = 12

This step clears the rational expressions, leaving a polynomial equation.

3

Expand and simplify the polynomial equation.

x^2 + 2x - 3x + 12 = 12

, which simplifies to

x^2 - x = 0

.

Expanding the terms allows us to combine like terms and set the quadratic equation to zero.

4

Factor the quadratic equation to solve for

x

.

x(x - 1) = 0

, giving candidate solutions

x = 0

or

x = 1

.

Applying the zero-product property identifies the roots of the quadratic equation.

5

Verify the candidate solutions against the initial restrictions.

Since

x = 0

makes the denominators in the original equation equal to zero, it is extraneous. The candidate solution

x = 1

is valid.

We must verify solutions because multiplying by a variable expression can introduce extraneous roots that make the original expressions undefined.

Key Concept

Solving rational equations by clearing denominators and checking for extraneous solutions.

Estimated Time:1m 30s

Question 307Question →

The daily profit, in dollars, of a small company is modeled by the quadratic function $P(x) = -2x^2 + 120x - 1000$ , where $x$ represents the number of items the company produces and sells each day. For what number of items produced and sold, greater than $20$ , will the company break even (meaning its daily profit is $\$0$ )?

10

30

40

50

Show answer & explanation

Answer: 50

Answer

The company breaks even when it produces and sells 50 items.

To find the break-even points, set the profit function to zero:

-2x^2 + 120x - 1000 = 0

. Dividing both sides by

-2

yields

x^2 - 60x + 500 = 0

. Factoring this equation gives

(x - 10)(x - 50) = 0

, which results in the solutions

x = 10

and

x = 50

. Since the question specifies that the number of items must be greater than

20

, the correct answer is 50.

Step-by-Step Solution

1

Set the profit function equal to zero to find the break-even points.

-2x^2 + 120x - 1000 = 0

Breaking even means the profit,

P(x)

, is equal to 0.

2

Divide the entire equation by the common factor of

-2

to simplify the quadratic expression.

x^2 - 60x + 500 = 0

Simplifying the quadratic expression makes it easier to factor.

3

Factor the quadratic equation by finding two numbers that multiply to

500

and add to

-60

.

(x - 10)(x - 50) = 0

Factoring allows us to find the individual roots of the equation.

4

Solve for

x

by setting each factor equal to zero.

x = 10

or

x = 50

The zero product property states that if a product of factors is zero, at least one factor must be zero.

5

Apply the constraint that the number of items must be greater than

20

.

x = 50

Out of the two roots, only 50 is greater than 20.

Key Concept

Solving quadratic equations in real-world contexts by finding roots and applying constraints.

Question 308Question →

In the quadratic equation $2x^2 - 12x + k = 0$ , $k$ is a constant. If the sum of the squares of the solutions to the equation is $26$ , what is the value of $k$ ?

Show answer & explanation

Answer: 10

Answer

The value of

k

is

10

.

The correct answer is

10

. By expressing the sum of the squares of the solutions as

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

, we can substitute the sum of the solutions (

-\frac{-12}{2} = 6

) and the product of the solutions (

\frac{k}{2}

) directly into the expression. This gives

26 = 36 - k

. Solving for

k

yields

k = 10

. Alternatively, solving the quadratic equation using the quadratic formula yields solutions

3 + \frac{\sqrt{144-8k}}{4}

and

3 - \frac{\sqrt{144-8k}}{4}

. Squaring these solutions and setting their sum equal to

26

simplifies to

18 + 2\left(\frac{144-8k}{16}\right) = 26

, which also solves to

k = 10

.

Step-by-Step Solution

1

Find the sum and product of the solutions using the coefficients of the quadratic equation.

The sum of the solutions is

6

and the product of the solutions is

\frac{k}{2}

.

By Vieta's formulas, for any quadratic equation

ax^2 + bx + c = 0

with solutions

x_1

and

x_2

, the sum of the solutions is

x_1 + x_2 = -\frac{b}{a}

and the product of the solutions is

x_1 x_2 = \frac{c}{a}

.

2

Apply the algebraic identity to express the sum of the squares of the solutions in terms of their sum and product.

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

This identity allows us to substitute the sum and product of the solutions directly without solving for the individual solutions first.

3

Substitute the values and solve for

k

.

26 = 6^2 - 2\left(\frac{k}{2}\right) \implies 26 = 36 - k \implies k = 10

Substituting the given sum of squares (

26

), the sum of solutions (

6

), and the product of solutions (

\frac{k}{2}

) allows us to solve for the unknown constant

k

directly.

Key Concept

Using the relationship between the roots and coefficients of a quadratic equation (Vieta's formulas) in combination with algebraic identities to solve for unknown constants.

Question 309Question →

If

x

satisfies the equation below, what is the value of

x - 1

?

\sqrt{30 - 2x} = x - 3

-4

2

6

8

Show answer & explanation

Answer: 6

Answer

6

The correct answer is

6

. Squaring both sides of the equation

\sqrt{30 - 2x} = x - 3

yields

30 - 2x = x^2 - 6x + 9

. Rearranging terms into standard quadratic form gives

x^2 - 4x - 21 = 0

, which factors as

(x - 7)(x + 3) = 0

. This gives potential solutions of

x = 7

and

x = -3

. Checking these in the original equation shows that

x = 7

is valid since

\sqrt{30 - 2(7)} = 7 - 3 \implies 4 = 4

, while

x = -3

is extraneous since

\sqrt{30 - 2(-3)} = -3 - 3 \implies 6 \neq -6

. Therefore, the value of the requested expression

x - 1

is

7 - 1 = 6

.

Step-by-Step Solution

1

Square both sides of the equation to eliminate the radical.

30 - 2x = (x - 3)^2

To solve a radical equation, squaring both sides isolates the terms under the square root.

2

Expand the right side and move all terms to one side to set the quadratic equation to zero.

x^2 - 4x - 21 = 0

Expanding

(x - 3)^2

yields

x^2 - 6x + 9

. Subtracting

30

and adding

2x

to both sides results in a standard quadratic form.

3

Factor the quadratic equation.

(x - 7)(x + 3) = 0

Factoring the quadratic helps find the potential solutions for

x

.

4

Identify potential solutions and substitute them back into the original equation to check for extraneous solutions.

x = 7

is the only valid solution;

x = -3

is extraneous.

Substituting

x = 7

gives

\sqrt{30 - 14} = 7 - 3

, which simplifies to

4 = 4

(true). Substituting

x = -3

gives

\sqrt{30 - 2(-3)} = -3 - 3

, which simplifies to

6 = -6

(false).

5

Calculate the value of the expression

x - 1

using the valid solution

x = 7

.

7 - 1 = 6

The question asks for the value of

x - 1

, so we substitute

7

for

x

.

Key Concept

Solving radical equations and identifying extraneous solutions.

Estimated Time:1m 35s

Question 310Question →

If $4^{a} \cdot 8^{b} = 32^{5}$ and $a + b = 9$ , what is the value of $b$ ?

2

5

7

11

Show answer & explanation

Answer: 7

Answer

7

By converting all bases to 2, the equation

4^a \cdot 8^b = 32^5

becomes

(2^2)^a \cdot (2^3)^b = (2^5)^5

, which simplifies to

2^{2a+3b} = 2^{25}

. Equating the exponents gives the linear equation

2a + 3b = 25

. Since we are given that

a + b = 9

, we can multiply this equation by 2 to get

2a + 2b = 18

. Subtracting the two equations yields

b = 7

.

Step-by-Step Solution

1

Express each base in the equation

4^a \cdot 8^b = 32^5

as a power of 2.

Since

4 = 2^2

,

8 = 2^3

, and

32 = 2^5

, the equation becomes

(2^2)^a \cdot (2^3)^b = (2^5)^5

.

To solve exponential equations with different bases, it is helpful to express all terms using a common base.

2

Apply the power-of-a-power property

(x^m)^n = x^{mn}

and the product-of-powers property

x^m \cdot x^n = x^{m+n}

to simplify both sides.

The left side simplifies to

2^{2a} \cdot 2^{3b} = 2^{2a+3b}

, and the right side simplifies to

2^{25}

. The equation is now

2^{2a+3b} = 2^{25}

.

Simplifying the expressions allows us to equate the exponents.

3

Set the exponents equal to each other to form a linear equation.

2a + 3b = 25

.

If two exponential expressions with the same base are equal, their exponents must be equal.

4

Solve the system of equations consisting of

2a + 3b = 25

and

a + b = 9

.

Multiplying the second equation by 2 gives

2a + 2b = 18

. Subtracting this from

2a + 3b = 25

yields

(2a + 3b) - (2a + 2b) = 25 - 18

, which simplifies to

b = 7

.

Eliminating one variable allows us to solve for the other variable directly.

Key Concept

Solving exponential equations by converting terms to a common base and solving the resulting system of linear equations.

Alternative Method

Substitute the answer options for

b

back into the equations. If

b = 7

, then

a = 9 - 7 = 2

. Plugging these values into the left side of the exponential equation gives

4^2 \cdot 8^7 = 16 \cdot (2^3)^7 = 2^4 \cdot 2^{21} = 2^{25}

. The right side is

32^5 = (2^5)^5 = 2^{25}

. Since both sides are equal, 7 is the correct answer.

Estimated Time:1m 30s

Question 311Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -x^2 + 6x + 7$ has a vertex at $(h, k)$ . If the graph of $f$ is translated $4$ units to the right and $3$ units up to produce the graph of the function $g$ , what is the maximum value of $g$ ?

Show answer & explanation

Answer: 19

Answer

The maximum value of the function

g

is

19

.

The vertex of the original quadratic function

f(x) = -x^2 + 6x + 7

is

(3, 16)

. Since the coefficient of

x^2

is negative, the graph opens downward, making

16

the maximum value of the function. Translating the graph

3

units up shifts all

y

-values up by

3

, which increases the maximum value to

16 + 3 = 19

. The horizontal translation of

4

units to the right shifts the graph horizontally but does not affect the maximum output value.

Step-by-Step Solution

1

Find the vertex of the original quadratic function

f(x) = -x^2 + 6x + 7

.

The vertex of

f(x)

is

(3, 16)

.

Rewrite the function in vertex form,

f(x) = -(x-3)^2 + 16

, to identify the vertex

(h, k)

as

(3, 16)

.

2

Determine the maximum value of the original function

f(x)

.

The maximum value of

f(x)

is

16

.

Since the leading coefficient is negative, the parabola opens downward, and the maximum value occurs at the vertex's

y

-coordinate.

3

Apply the vertical translation to find the maximum value of the new function

g(x)

.

The maximum value of

g

is

19

.

Translating the graph

3

units up increases all

y

-values, including the maximum value, by

3

, so

16 + 3 = 19

. The horizontal translation of

4

units to the right does not affect the maximum value.

Key Concept

Quadratic Functions and Graphs

Question 312Question →

The equation $\sqrt{2x + 12} - x = -6$ has one real solution. What is this solution?

Show answer & explanation

Answer: 12

Answer

12

Substituting 12 into the original equation

\sqrt{2x + 12} - x = -6

yields

\sqrt{2(12) + 12} - 12 = \sqrt{36} - 12 = 6 - 12 = -6

. Since this creates a true statement, 12 is the unique real solution to the equation.

Step-by-Step Solution

1

Isolate the radical term by adding

x

to both sides of the equation.

\sqrt{2x + 12} = x - 6

Isolating the radical on one side allows us to eliminate it by squaring both sides in the next step.

2

Square both sides of the equation to clear the square root.

2x + 12 = (x - 6)^2

Squaring a square root cancels the radical, allowing us to solve the equation algebraically.

3

Expand the squared binomial on the right-hand side.

2x + 12 = x^2 - 12x + 36

Expanding

(x - 6)^2

to

x^2 - 12x + 36

helps set up a standard quadratic form.

4

Subtract

2x

and

12

from both sides to form a quadratic equation equal to zero.

x^2 - 14x + 24 = 0

A quadratic equation must be in the form

ax^2 + bx + c = 0

to solve by factoring or the quadratic formula.

5

Factor the quadratic equation.

(x - 12)(x - 2) = 0

Factoring finds two numbers that multiply to 24 and add to -14, which are -12 and -2.

6

Test the potential solutions

x = 12

and

x = 2

in the original equation to identify any extraneous solutions.

For

x = 2

:

\sqrt{2(2) + 12} - 2 = \sqrt{16} - 2 = 4 - 2 = 2 \neq -6

. For

x = 12

:

\sqrt{2(12) + 12} - 12 = \sqrt{36} - 12 = 6 - 12 = -6

. Thus,

x = 12

is the only valid solution.

Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original equation.

Key Concept

Solving radical equations by isolating the radical, squaring both sides, solving the resulting quadratic equation, and verifying all potential solutions to check for extraneous values.

Estimated Time:1m 30s

Question 313Question →

The function $f$ is defined by $f(x) = x^3 - 3x^2 - 10x + k$ , where $k$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ has $x$ -intercepts at $(c, 0)$ and $(2c, 0)$ , where $c$ is a positive constant. What is the value of $k$ ?

Show answer & explanation

Answer: 24

Answer

24

By applying the factor theorem to the two

x

-intercepts

(c,0)

and

(2c,0)

, we establish the system

c^3 - 3c^2 - 10c + k = 0

and

8c^3 - 12c^2 - 20c + k = 0

. Subtracting these equations yields

7c^3 - 9c^2 - 10c = 0

. Since

c > 0

, dividing by

c

gives

7c^2 - 9c - 10 = 0

, which factors as

(7c+5)(c-2)=0

, yielding the positive solution

c=2

. Substituting

c=2

back into the first equation gives

8 - 12 - 20 + k = 0

, which solves to

k = 24

.

Step-by-Step Solution

1

Set up equations for the roots

c

and

2c

using the factor theorem.

f(c) = c^3 - 3c^2 - 10c + k = 0

and

f(2c) = 8c^3 - 12c^2 - 20c + k = 0

An

x

-intercept at

(r, 0)

means that

r

is a root of the polynomial, so

f(r) = 0

.

2

Subtract the first equation from the second to eliminate the constant

k

.

7c^3 - 9c^2 - 10c = 0

Eliminating

k

allows us to solve for the root

c

directly.

3

Solve the polynomial equation for the positive constant

c

.

c = 2

Dividing the equation by

c

(since

c > 0

) yields

7c^2 - 9c - 10 = 0

, which factors as

(7c+5)(c-2) = 0

. Since

c

must be positive,

c = 2

.

4

Substitute

c = 2

back into the equation for

f(c) = 0

to solve for

k

.

k = 24

Substituting the known root value allows us to find the value of the constant coefficient

k

.

Key Concept

The relationship between a polynomial's algebraic factors, its roots, and its

x

-intercepts in the coordinate plane.

Estimated Time:2m 0s

Question 314Question →

In the $xy$ -plane, the graph of the function $g$ is obtained by applying a sequence of transformations to the graph of the function $f(x) = |x + 2| - 3$ . Specifically, the graph of $g$ is a vertical stretch and translation of the graph of $f$ , such that $g(x) = a f(x - h) + k$ for some constants $a$ , $h$ , and $k$ . The vertex of the graph of $g$ is located at $(1, 5)$ , and the graph of $g$ passes through the point $(0, -1)$ . What is the value of $g(3)$ ?

-5

-7

17

-43

Show answer & explanation

Answer: -7

Answer

The value of

g(3)

is

-7

.

The correct value of

-7

is obtained by first identifying the vertex of

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

at

(-2, -3)

. Comparing this to the vertex of

g(x)

at

(1, 5)

yields the horizontal shift parameter

h = 3

and the equation

-3a + k = 5

. Using the point

(0, -1)

yields the second equation

-2a + k = -1

. Solving this system gives

a = -6

and

k = -13

. Substituting these into the formula for

g(3)

yields

-7

.

Step-by-Step Solution

1

Identify the vertex of the function

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

.

The vertex of

f(x)

is at

(-2, -3)

.

The vertex of an absolute value function of the form

y = |x - x_0| + y_0

is located at

(x_0, y_0)

.

2

Relate the vertex of

f(x)

to the vertex of

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

.

h = 3

and

-3a + k = 5

.

The horizontal shift

h

moves the vertex from

x = -2

to

x = 1

, so

-2 + h = 1 \implies h = 3

. The vertical stretch and translation transform the

y

-coordinate of the vertex from

-3

to

5

, so

a(-3) + k = 5

.

3

Use the given point

(0, -1)

to set up a second equation.

-2a + k = -1

.

Since the graph of

g(x)

passes through

(0, -1)

, we evaluate

g(0) = a f(0 - 3) + k = -1

. Evaluating

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

yields the equation

-2a + k = -1

.

4

Solve the system of equations for

a

and

k

.

a = -6

and

k = -13

.

Subtracting

-3a + k = 5

from

-2a + k = -1

gives

a = -6

. Substituting

a = -6

back into either equation yields

k = -13

.

5

Evaluate

g(3)

using the completed function formula

g(x) = -6 f(x - 3) - 13

.

g(3) = -7

.

We substitute

x = 3

into the equation to get

g(3) = -6 f(0) - 13

. Evaluating

f(0) = |0 + 2| - 3 = -1

gives

g(3) = -6(-1) - 13 = 6 - 13 = -7

.

Key Concept

Function transformations including horizontal translations, vertical translations, and vertical scaling.

Question 315Question →

The graph of the quadratic function $f(x) = -2(x - 3)^2 + a$ in the $xy$ -plane has a $y$ -intercept at $(0, -10)$ , where $a$ is a constant. What is the maximum value of $f(x)$ ?

-28

-10

3

8

Show answer & explanation

Answer: $8$

Answer

The maximum value of the function is

8

.

The correct answer is

8

. Since the

y

-intercept of the graph is

(0, -10)

, we substitute

x = 0

into the function:

f(0) = -2(0 - 3)^2 + a = -10

. Simplifying this yields

-2(9) + a = -10

, which becomes

-18 + a = -10

. Adding

18

to both sides gives

a = 8

. The equation of the function is therefore

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

. Since this is in vertex form, the vertex is

(3, 8)

. Because the coefficient of the squared term is negative, the parabola opens downward, and the maximum value of the function is the

y

-coordinate of the vertex, which is

8

.

Step-by-Step Solution

1

Use the given

y

-intercept to set up an equation for the constant

a

.

f(0) = -10 \implies -2(0 - 3)^2 + a = -10

The

y

-intercept occurs where the input

x

is equal to

0

.

2

Simplify the expression and solve for

a

.

-2(-3)^2 + a = -10 \implies -2(9) + a = -10 \implies -18 + a = -10 \implies a = 8

Squaring

-3

yields

9

, and multiplying by

-2

gives

-18

. Adding

18

to both sides isolates

a

.

3

Identify the vertex of the quadratic function and determine the maximum value.

The function is

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

. The vertex of this parabola is

(3, 8)

. Since the leading coefficient

-2

is negative, the parabola opens downward, meaning the

y

-coordinate of the vertex,

8

, is the maximum value of

f(x)

.

A quadratic function in vertex form

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

has its vertex at

(h, k)

. If

a < 0

, the maximum value is

k

.

Key Concept

Identifying the vertex and maximum value of a quadratic function from its vertex form and

y

-intercept.

Question 316Question →

If $\frac{4^{x+3}}{8^{x-1}} = 16^{3-x}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 1

Answer

1

The correct answer is 1. By writing all terms with a base of 2, the equation is expressed as

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

. Applying the power rule to simplify the exponents yields

\frac{2^{2x+6}}{2^{3x-3}} = 2^{12-4x}

. Using the quotient rule, we subtract the exponents on the left-hand side to get

2^{(2x+6)-(3x-3)} = 2^{-x+9}

. Equating the exponents gives the linear equation

-x+9 = 12-4x

. Adding

4x

and subtracting

9

from both sides results in

3x = 3

, which simplifies to

x=1

.

Step-by-Step Solution

1

Express all exponential terms using a common base of 2

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

Rewriting each base as a power of 2 allows the application of standard exponent rules.

2

Apply the power of a power rule

(a^m)^n = a^{mn}

to distribute the exponents

\frac{2^{2x+6}}{2^{3x-3}} = 2^{12-4x}

Multiplying the inner exponent by each term of the outer exponent simplifies the expression.

3

Apply the quotient of powers rule

\frac{a^m}{a^n} = a^{m-n}

to combine the fraction

2^{-x+9} = 2^{12-4x}

Subtracting the exponent in the denominator from the exponent in the numerator simplifies the left-hand side.

4

Equate the exponents since the bases are equal

-x + 9 = 12 - 4x

If two exponential expressions with the same positive base are equal, their exponents must be equal.

5

Solve the linear equation for

x

x = 1

Isolating

x

by algebraic manipulation yields the final answer.

Key Concept

Solving exponential equations by finding a common base and applying the laws of exponents.

Question 317Question →

A function $f$ has exactly two local extrema: a local maximum at the point $(-2, 5)$ and a local minimum at the point $(2, -3)$ . A second function $g$ is defined by $g(x) = 1 - 3f(2x - 4)$ . If the local minimum of the graph of $y = g(x)$ occurs at the point $(h, k)$ , what is the value of $h + k$ ?

-17

-15

-13

-5

Show answer & explanation

Answer: -13

Answer

-13

To locate the local minimum of the transformed function

g(x) = 1 - 3f(2x - 4)

, we analyze the vertical reflection and scaling. The negative coefficient in

-3f(2x - 4)

reflects the graph vertically, which means the local maximum of

f(x)

at the point

(-2, 5)

becomes the local minimum of

g(x)

. To find the corresponding

x

-coordinate

h

, we solve

2h - 4 = -2

, which gives

h = 1

. To find the corresponding

y

-coordinate

k

, we evaluate

g(1) = 1 - 3f(-2) = 1 - 3(5) = -14

. Thus, the local minimum occurs at

(1, -14)

, and the sum of these coordinates is

1 + (-14) = -13

.

Step-by-Step Solution

1

Analyze how the vertical reflection in

g(x) = 1 - 3f(2x - 4)

affects the extrema.

Due to the negative coefficient in

-3f(2x - 4)

, the graph is reflected vertically. Therefore, the local maximum of

f(x)

at

(-2, 5)

transforms into the local minimum of

g(x)

, while the local minimum of

f(x)

transforms into the local maximum of

g(x)

.

A vertical reflection inverts the relative heights of the outputs, converting peaks to valleys and vice versa.

2

Determine the horizontal transformation to find the

x

-coordinate

h

of the new local minimum.

Set the input of

f

in the definition of

g(x)

equal to the

x

-coordinate of the maximum of

f(x)

, which is

-2

:

2h - 4 = -2

. Solving this equation gives

2h = 2

, which yields

h = 1

.

The horizontal shift and compression require solving for the new input variable that produces the same argument for the inner function.

3

Determine the vertical transformation to find the

y

-coordinate

k

of the new local minimum.

Substitute

h = 1

into

g(x)

to find the output value:

k = g(1) = 1 - 3f(2(1) - 4) = 1 - 3f(-2)

. Since the maximum value of

f(x)

is

f(-2) = 5

, we compute

k = 1 - 3(5) = 1 - 15 = -14

.

The vertical transformations (stretch, reflection, and shift) are applied directly to the function output.

4

Calculate the sum of the coordinates

h + k

.

Compute

h + k = 1 + (-14) = -13

.

The question asks for the sum of the coordinates of the local minimum of

g(x)

.

Key Concept

Analyzing function transformations including horizontal compression, horizontal translation, vertical stretch, reflection, and vertical translation to determine the coordinates of key features (local extrema) of a transformed function.

Estimated Time:3m 0s

Question 318Question →

In the quadratic equation $2x^2 - bx + 18 = 0$ , $b$ is a positive constant. If one of the solutions to the equation is $4$ times the other solution, what is the value of $b$ ?

Show answer & explanation

Answer: 15

Answer

15

By Vieta's formulas, the product of the roots of the quadratic equation

2x^2 - bx + 18 = 0

is

\frac{18}{2} = 9

. Letting the roots be

r_1

and

r_2

with

r_1 = 4r_2

, we have

4r_2^2 = 9

, which yields

r_2 = \frac{3}{2}

(since

b > 0

implies the roots must be positive). Thus,

r_1 = 6

. The sum of the roots is

6 + \frac{3}{2} = \frac{15}{2}

, and by Vieta's formulas, this sum equals

\frac{b}{2}

. Solving for

b

gives

15

.

Step-by-Step Solution

1

Set up the relationships for the product and sum of the roots using Vieta's formulas.

r_1 \cdot r_2 = 9

and

r_1 + r_2 = \frac{b}{2}

Vieta's formulas relate the coefficients of a quadratic equation to the sum and product of its roots.

2

Substitute the given condition that one root is

4

times the other (

r_1 = 4r_2

) into the product equation.

4r_2^2 = 9

This reduces the product equation to a single variable equation in terms of

r_2

.

3

Solve for

r_2

and determine its sign based on the constraint that

b

is positive.

r_2 = \frac{3}{2}

Since

b > 0

, the sum of the roots

5r_2 = \frac{b}{2}

must be positive, which requires

r_2 > 0

.

4

Calculate the second root

r_1

and then use the sum of the roots to find

b

.

b = 15

The sum of the roots is

6 + \frac{3}{2} = \frac{15}{2}

, and since

r_1 + r_2 = \frac{b}{2}

, we have

\frac{b}{2} = \frac{15}{2}

.

Key Concept

Relationship between roots and coefficients of a quadratic equation

Question 319Question →

In the quadratic equation $x^2 - px + q = 0$ , $p$ and $q$ are positive constants. If the equation has exactly one real solution, what is the value of $\frac{p^2}{q}$ ?

-4

4

2

16

Show answer & explanation

Answer: $4$

Answer

The correct answer is

4

.

The correct answer is

4

. For any quadratic equation of the form

ax^2 + bx + c = 0

to have exactly one real solution, its discriminant

\Delta = b^2 - 4ac

must equal

0

. In the equation

x^2 - px + q = 0

, we identify

a = 1

,

b = -p

, and

c = q

. Substituting these values into the discriminant formula gives

(-p)^2 - 4(1)(q) = 0

, which simplifies to

p^2 - 4q = 0

. Rearranging this equation gives

p^2 = 4q

. Since

q

is a positive constant, we can divide both sides of the equation by

q

to isolate the ratio, yielding

\frac{p^2}{q} = 4

.

Step-by-Step Solution

1

Recall the condition for a quadratic equation of the form

ax^2 + bx + c = 0

to have exactly one real solution.

The discriminant,

\Delta = b^2 - 4ac

, must be equal to

0

.

A quadratic equation has exactly one real solution if and only if its discriminant is zero, meaning the parabola is tangent to the x-axis.

2

Identify the coefficients of the given quadratic equation

x^2 - px + q = 0

and substitute them into the discriminant formula.

Here,

a = 1

,

b = -p

, and

c = q

. Thus,

\Delta = (-p)^2 - 4(1)(q) = p^2 - 4q = 0

.

To apply the one-solution condition specifically to the coefficients of the given equation.

3

Solve the equation

p^2 - 4q = 0

for the ratio

\frac{p^2}{q}

.

p^2 = 4q \implies \frac{p^2}{q} = 4

.

Isolate the target expression by dividing both sides of the equation by

q

, which is valid since

q

is a positive constant and thus non-zero.

Key Concept

Evaluating the discriminant of a quadratic equation to determine the number of real solutions.

Estimated Time:1m 30s

Question 320Question →

A quadratic function $f$ has a vertex at $(h, k)$ , where $h$ and $k$ are constants. In the $xy$ -plane, the graph of $y = f(x)$ contains the points $(-1, 3)$ and $(7, 3)$ . The function $g$ is defined by $g(x) = f(x + 2) - 4$ , and its graph has a vertex at $(p, q)$ . If $p + q = 5$ , what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

The value of k is 8.

The axis of symmetry of the quadratic function f is located at the midpoint of the x-values of any two points on the graph that have the same y-value. Since the graph of f contains (-1, 3) and (7, 3), the x-coordinate of its vertex, h, is equal to the midpoint (-1 + 7) / 2 = 3. Thus, the vertex of f is (3, k). The definition of g(x) = f(x + 2) - 4 indicates a horizontal translation of 2 units to the left and a vertical translation of 4 units down. Applying these translations to the vertex (3, k) of f gives the vertex of g as (3 - 2, k - 4) = (1, k - 4). Since the vertex of g is defined as (p, q), we have p = 1 and q = k - 4. Substituting these values into the equation p + q = 5 yields 1 + (k - 4) = 5, which simplifies to k - 3 = 5, or k = 8.

Step-by-Step Solution

1

Find the x-coordinate of the vertex of the function f

h = 3

Because the points (-1, 3) and (7, 3) have the same y-coordinate, the axis of symmetry (and thus the x-coordinate of the vertex) must lie exactly halfway between x = -1 and x = 7.

2

Determine the vertex of the function g(x)

(p, q) = (1, k - 4)

The function g(x) = f(x + 2) - 4 represents a translation of the graph of f by 2 units to the left and 4 units down. Since the vertex of f is (3, k), the vertex of g is (3 - 2, k - 4).

3

Solve for k using the given sum p + q = 5

k = 8

Substitute p = 1 and q = k - 4 into the equation p + q = 5 to get 1 + k - 4 = 5, which simplifies to k = 8.

Key Concept

Quadratic functions and graphs, including symmetry and translations.

Advanced Math

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