Advanced Math Practice Questions — SAT — Page 17

Question 321Question →

In the $xy$ -plane, the graph of the cubic function $f(x) = x^3 - 7x^2 + kx - 12$ , where $k$ is a constant, is tangent to the $x$ -axis at one point and intersects the $x$ -axis at another point. If all roots of $f(x)$ are real numbers, which of the following could be the value of $k$ ?

16

-16

7

12

Show answer & explanation

Answer: 16

Answer

16

The correct answer is the value that makes the cubic function have a double root at

2

and a single root at

3

. Setting

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

and expanding it yields

x^3 - 7x^2 + 16x - 12

. Comparing this to

f(x) = x^3 - 7x^2 + kx - 12

shows that

k = 16

.

Step-by-Step Solution

1

Express the cubic function in factored form based on its roots.

f(x) = (x - r_1)^2(x - r_2)

Since the graph is tangent to the

x

-axis at one point and intersects it at another, it has one real root of multiplicity 2 (let it be

r_1

) and one real root of multiplicity 1 (let it be

r_2

).

2

Expand the factored form to obtain coefficients in terms of

r_1

and

r_2

.

f(x) = x^3 - (2r_1 + r_2)x^2 + (r_1^2 + 2r_1 r_2)x - r_1^2 r_2

Comparing this expansion to the given expression

f(x) = x^3 - 7x^2 + kx - 12

allows us to set up a system of equations.

3

Set up and solve the system of equations for

r_1

and

r_2

.

2r_1 + r_2 = 7

,

r_1^2 r_2 = 12

, and

k = r_1^2 + 2r_1 r_2

. Substituting

r_2 = 7 - 2r_1

into the second equation gives

2r_1^3 - 7r_1^2 + 12 = 0

. Testing integer root candidates shows

r_1 = 2

is a solution, which yields

r_2 = 3

.

This determines the values of the roots

r_1

and

r_2

.

4

Calculate the value of

k

using the values of the roots.

k = 2^2 + 2(2)(3) = 16

Substituting

r_1 = 2

and

r_2 = 3

into the expression for

k

gives the target coefficient.

Key Concept

Analyzing the relationship between a polynomial's algebraic factors, roots, and its graphical features such as tangency and intercepts.

Question 322Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -2(x - d)^2 + 8$ , where $d$ is a positive constant, intersects the $x$ -axis at the point $(2, 0)$ . The function $g$ is defined by $g(x) = f(x + 3) - 5$ . What is the $y$ -coordinate of the $y$ -intercept of the graph of $g$ in the $xy$ -plane?

-15

-95

1

11

Show answer & explanation

Answer: 1

Answer

The

y

-coordinate of the

y

-intercept of the graph of

g

is

1

.

To find the correct answer, we first substitute the point

(2,0)

into

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

to obtain

0 = -2(2 - d)^2 + 8

. Simplifying this equation gives

(2 - d)^2 = 4

, which has solutions

d = 0

and

d = 4

. Since the problem states that

d

is a positive constant, we have

d = 4

, which means

f(x) = -2(x - 4)^2 + 8

. We then find the equation for the transformed function

g(x)

by replacing

x

with

x + 3

and subtracting

5

, which yields

g(x) = f(x + 3) - 5 = -2(x + 3 - 4)^2 + 8 - 5 = -2(x - 1)^2 + 3

. To find the

y

-coordinate of the

y

-intercept, we evaluate

g(x)

at

x = 0

, giving

g(0) = -2(0 - 1)^2 + 3 = -2(1) + 3 = 1

.

Step-by-Step Solution

1

Use the given

x

-intercept

(2,0)

of

f(x)

to solve for the constant

d

.

d = 4

Substitute

x = 2

and

f(x) = 0

into the equation

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

to get

0 = -2(2 - d)^2 + 8

. Dividing by

-2

and isolating the squared term gives

(2 - d)^2 = 4

. Taking the square root of both sides gives

2 - d = 2

or

2 - d = -2

, which yields

d = 0

or

d = 4

. Since

d

must be a positive constant, we must choose

d = 4

.

2

Determine the equation of the transformed function

g(x)

.

g(x) = -2(x - 1)^2 + 3

Substitute

d = 4

into

f(x)

to get

f(x) = -2(x - 4)^2 + 8

. Since

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

, substitute

x + 3

for

x

in

f(x)

and subtract

5

:

g(x) = -2((x + 3) - 4)^2 + 8 - 5

, which simplifies to

g(x) = -2(x - 1)^2 + 3

.

3

Find the

y

-coordinate of the

y

-intercept of the graph of

g

.

1

The

y

-intercept of a graph is the point where

x = 0

. Substitute

x = 0

into the equation for

g(x)

to find

g(0) = -2(0 - 1)^2 + 3 = -2(1) + 3 = 1

.

Key Concept

Applying horizontal and vertical translations to quadratic equations and identifying graph features.

Question 323Question →

An equation is shown below.

\sqrt{2x + 7} = x - 4

If

x

satisfies the equation above, what is the value of

x + 3

?

4

2

12

6

Show answer & explanation

Answer: 12

Answer

12

To solve the equation, square both sides to obtain

2x + 7 = x^2 - 8x + 16

. Rearranging terms gives the quadratic equation

x^2 - 10x + 9 = 0

, which factors as

(x - 9)(x - 1) = 0

. This yields potential solutions of

x = 9

and

x = 1

. Testing these in the original equation shows that

x = 9

is a valid solution because

\sqrt{25} = 5

, but

x = 1

is extraneous because

\sqrt{9} \neq -3

. Thus, the only real solution is

x = 9

. Evaluating

x + 3

for this solution gives

9 + 3 = 12

.

Step-by-Step Solution

1

Square both sides of the equation to eliminate the radical.

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

, which expands to

2x + 7 = x^2 - 8x + 16

.

Squaring both sides is the standard method to solve radical equations.

2

Rearrange the terms to set the quadratic equation to zero.

x^2 - 10x + 9 = 0

.

This allows us to solve the quadratic equation by factoring.

3

Factor the quadratic equation.

(x - 9)(x - 1) = 0

, giving potential solutions of

x = 9

and

x = 1

.

Factoring helps find the roots of the quadratic equation.

4

Verify each potential solution in the original equation to check for extraneous roots.

x = 9

is valid because

\sqrt{2(9)+7} = 9-4

is true (

5=5

).

x = 1

is extraneous because

\sqrt{2(1)+7} = 1-4

is false (

3 \neq -3

).

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

5

Substitute the valid solution

x = 9

into the expression

x + 3

.

9 + 3 = 12

.

The question asks for the value of the expression

x + 3

rather than just

x

.

Key Concept

Solving radical equations and checking for extraneous solutions

Question 324Question →

A sample of a radioactive isotope decays according to the function $A(t) = A_0 \left(\frac{1}{2}\right)^{\frac{t}{h}}$ , where $A_0$ is the initial mass of the sample, $t$ is the time in years, and $h$ is the half-life of the isotope in years. If the sample decays to $12.5\%$ of its initial mass in $45$ years, what is the value of $h$ ?

5.625

22.5

15

135

Show answer & explanation

Answer: 15

Answer

15

The correct answer is 15. The remaining fraction of the radioactive isotope is

12.5\%

, which can be written as

0.125

or

\frac{1}{8}

. Expressing this as a power of the decay base yields

\frac{1}{8} = \left(\frac{1}{2}\right)^3

. Setting the decay formula equal to this fraction gives

\left(\frac{1}{2}\right)^{\frac{45}{h}} = \left(\frac{1}{2}\right)^3

. Equating the exponents gives

\frac{45}{h} = 3

, which yields

h = 15

.

Step-by-Step Solution

1

Set up the decay equation based on the given remaining percentage.

A(45) = 0.125 A_0

, which simplifies to

\frac{A(45)}{A_0} = 0.125 = \frac{1}{8}

.

This establishes the fraction of the substance remaining after

45

years.

2

Substitute the remaining fraction and the time

t = 45

into the exponential decay model.

\left(\frac{1}{2}\right)^{\frac{45}{h}} = \frac{1}{8}

.

This sets up the equation in terms of the unknown half-life parameter

h

.

3

Express both sides of the equation with a common base of

\frac{1}{2}

.

\left(\frac{1}{2}\right)^{\frac{45}{h}} = \left(\frac{1}{2}\right)^3

.

Converting

\frac{1}{8}

to

\left(\frac{1}{2}\right)^3

allows us to equate the exponents directly.

4

Equate the exponents and solve for

h

.

\frac{45}{h} = 3 \implies 3h = 45 \implies h = 15

.

Since the bases are identical, their exponents must be equal to satisfy the equation.

Key Concept

Solving exponential equations by expressing both sides with a common base.

Estimated Time:1m 30s

Question 325Question →

A quadratic function $f$ is defined by $f(x) = a(x - 4)(x - 10)$ , where $a$ is a positive constant. In the $xy$ -plane, the graph of $f$ has a vertex with a $y$ -coordinate of $-18$ . What is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

The value of

a

is 2.

The vertex of a parabola lies on the axis of symmetry, which is located midway between its x-intercepts. Since the function is

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

, the x-intercepts are at

x = 4

and

x = 10

. The midpoint of these values is

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

. Thus, the x-coordinate of the vertex is 7. Since the y-coordinate of the vertex is given as

-18

, the vertex is at

(7, -18)

. Substituting these coordinates into the function equation yields

-18 = a(7 - 4)(7 - 10)

, which simplifies to

-18 = a(3)(-3)

, or

-18 = -9a

. Solving for

a

gives

a = 2

.

Step-by-Step Solution

1

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

The x-coordinate of the vertex is 7.

The x-coordinate of the vertex lies on the axis of symmetry, which is the midpoint of the x-intercepts

x = 4

and

x = 10

.

2

Identify the coordinates of the vertex.

The vertex is at

(7, -18)

.

The y-coordinate of the vertex is given as

-18

, and its x-coordinate was calculated as 7.

3

Substitute the vertex coordinates into the function definition to solve for

a

.

a = 2

Substituting

(7, -18)

into

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

gives

-18 = a(7 - 4)(7 - 10)

, which simplifies to

-18 = -9a

.

Key Concept

Finding the vertex of a quadratic function from its factored form and solving for a leading coefficient.

Alternative Method

Alternatively, the function

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

can be expanded to standard form:

f(x) = a(x^2 - 14x + 40)

. The x-coordinate of the vertex can be found using the formula

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

, which gives

x = -\frac{-14a}{2a} = 7

. Then, substitute

x = 7

and

f(7) = -18

to solve for

a

.

Estimated Time:1m 30s

Question 326Question →

A quadratic function $f$ has its vertex at $(4, -3)$ in the coordinate plane. The point $(2, 5)$ is on the graph of $y = f(x)$ . The function $g$ is defined by $g(x) = -f(x - 3) + 8$ . What is the value of $g(8)$ ?

Show answer & explanation

Answer: 9

Answer

The correct answer is 9.

The correct answer is 9. To find this, we first establish the vertex form of the quadratic function

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

. Substituting the point

(2, 5)

gives

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

, which simplifies to

8 = 4a

, so

a = 2

. Thus,

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

. To find the value of

g(8)

, we substitute

8

into the definition of

g(x)

, yielding

g(8) = -f(8 - 3) + 8 = -f(5) + 8

. Evaluating

f(5)

gives

2(5 - 4)^2 - 3 = -1

. Finally, substituting

-1

back into the expression for

g(8)

gives

-(-1) + 8 = 9

.

Step-by-Step Solution

1

Write the quadratic function

f(x)

in vertex form using its vertex

(4, -3)

.

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

The vertex form of a quadratic function with vertex

(h, k)

is

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

.

2

Substitute the coordinates of the point

(2, 5)

into the vertex form to find the value of the constant

a

.

a = 2

, so

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

Since the point

(2, 5)

lies on the graph of

f

, substituting

x = 2

and

f(2) = 5

allows us to solve for the vertical stretch factor

a

.

3

Use the definition of

g(x)

to express

g(8)

in terms of

f

.

g(8) = -f(5) + 8

Substituting

x = 8

into the equation

g(x) = -f(x - 3) + 8

yields

g(8) = -f(8 - 3) + 8 = -f(5) + 8

.

4

Evaluate

f(5)

using the equation for

f(x)

found in Step 2.

f(5) = -1

Substituting

x = 5

into

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

gives

2(5 - 4)^2 - 3 = 2(1) - 3 = -1

.

5

Substitute the value of

f(5)

into the expression for

g(8)

to find the final result.

g(8) = 9

Substituting

f(5) = -1

into

g(8) = -f(5) + 8

yields

-(-1) + 8 = 1 + 8 = 9

.

Key Concept

Finding the equation of a quadratic function from its vertex and a point, and evaluating transformed functions using function notation.

Question 327Question →

In the equation $\frac{25^x}{5^{x-3}} = 125^2$ , what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

By writing all terms with a base of 5, the equation

\frac{25^x}{5^{x-3}} = 125^2

becomes

\frac{5^{2x}}{5^{x-3}} = 5^6

. Applying the quotient rule of exponents, the left side simplifies to

5^{2x - (x-3)} = 5^{x+3}

. Setting the exponents equal gives

x+3 = 6

, which yields

x = 3

.

Step-by-Step Solution

1

Express all terms with a common base of 5.

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

and

125^2 = (5^3)^2 = 5^6

.

Expressing all exponential terms with the same base allows the exponents to be equated directly once simplified.

2

Substitute these expressions back into the original equation and simplify the left side using the quotient rule of exponents.

\frac{5^{2x}}{5^{x-3}} = 5^{2x - (x-3)} = 5^{x+3}

. The equation becomes

5^{x+3} = 5^6

.

The quotient rule states that dividing exponential terms with the same base requires subtracting the exponent of the denominator from the exponent of the numerator:

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

.

3

Equate the exponents and solve for

x

.

x + 3 = 6 \implies x = 3

.

Since the bases on both sides of the equation are equal, their exponents must also be equal: if

b^y = b^z

where

b > 0

and

b \neq 1

, then

y = z

.

Key Concept

Solving exponential equations by expressing terms with a common base and applying exponent rules.

Estimated Time:1m 30s

Question 328Question →

An equation is shown below.

\frac{x+1}{x-2} + \frac{1}{x} = \frac{6}{x^2-2x}

If

x

satisfies the equation above, what is the value of

x

?

-2

2

-4

4

Show answer & explanation

Answer: -4

Answer

-4

The correct answer is

-4

. To solve the rational equation, we first identify the least common denominator as

x(x-2)

. Multiplying both sides by this expression eliminates the denominators, yielding

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

. Expanding and simplifying gives the quadratic equation

x^2 + 2x - 8 = 0

, which factors as

(x+4)(x-2) = 0

. This yields potential solutions of

x = -4

and

x = 2

. However, substituting

x = 2

back into the original equation results in division by zero, making

x = 2

extraneous. The only valid solution is

-4

.

Step-by-Step Solution

1

Find the least common denominator (LCD) of the rational expressions in the equation.

The denominators are

x-2

,

x

, and

x^2-2x

. Since

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

, the LCD is

x(x-2)

, with the restriction that

x \neq 0

and

x \neq 2

.

Finding a common denominator allows us to clear the fractions by multiplying both sides of the equation.

2

Multiply each term of the equation by the LCD to eliminate the denominators.

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

This clears the denominators and converts the rational equation into a polynomial equation.

3

Expand and simplify the resulting equation.

x^2 + x + x - 2 = 6

, which simplifies to

x^2 + 2x - 2 = 6

.

Combining like terms prepares the equation to be written in standard quadratic form.

4

Rearrange the terms into standard quadratic form:

ax^2 + bx + c = 0

.

x^2 + 2x - 8 = 0

Subtracting

6

from both sides sets the quadratic equation to zero so it can be solved by factoring.

5

Factor the quadratic equation.

(x+4)(x-2) = 0

Finding two numbers that multiply to

-8

and add to

2

(

4

and

-2

) allows us to solve for

x

.

6

Find the potential solutions by setting each factor to zero.

x = -4

or

x = 2

Applying the zero product property gives the potential roots of the quadratic equation.

7

Check the potential solutions in the original equation to identify any extraneous solutions.

For

x = -4

, the equation is defined and valid. For

x = 2

, the denominators

x-2

and

x^2-2x

become zero, which is undefined.

Rational equations can produce extraneous solutions that make the original denominators zero, so they must be checked and discarded.

Key Concept

Solving rational equations by finding a common denominator and checking for extraneous solutions.

Question 329Question →

A quadratic function $g$ is defined by $g(x) = 2x^2 - 12x + k$ , where $k$ is a constant. In the $xy$ -plane, the graph of $g$ has its vertex on the line $y = -5$ . What is the value of $k$ ?

-59

-23

-5

13

Show answer & explanation

Answer: 13

Answer

13

The

x

-coordinate of the vertex for the quadratic function

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

is found using the formula

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

. Substituting

a = 2

and

b = -12

yields

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

. Evaluating

g(3)

gives the

y

-coordinate of the vertex:

g(3) = 2(3)^2 - 12(3) + k = -18 + k

. Since the vertex lies on the line

y = -5

, setting

-18 + k = -5

and solving for

k

gives

k = 13

.

Step-by-Step Solution

1

Find the

x

-coordinate of the vertex of the parabola.

The

x

-coordinate is

3

.

For a quadratic function

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

, the

x

-coordinate of the vertex is given by

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

. Here,

a = 2

and

b = -12

, so

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

.

2

Evaluate the function at

x = 3

to find the

y

-coordinate of the vertex in terms of

k

.

The

y

-coordinate is

-18 + k

.

Substituting

x = 3

into the function gives

g(3) = 2(3)^2 - 12(3) + k = 18 - 36 + k = -18 + k

.

3

Set the

y

-coordinate of the vertex equal to

-5

and solve for

k

.

k = 13

.

Since the vertex lies on the line

y = -5

, the

y

-coordinate of the vertex must be

-5

. Thus,

-18 + k = -5 \implies k = 13

.

Key Concept

Determining the vertex of a quadratic function and using its coordinates to solve for unknown parameters.

Question 330Question →

Several values of $x$ and the corresponding values of $f(x)$ for the quadratic function $f$ are shown in the table below.

$x$	$f(x)$
$-4$	$18$
$-2$	$6$
$0$	$2$
$2$	$6$
$4$	$18$

The function $g$ is defined by $g(x) = -2(f(x - 3) + 4)$ . If the vertex of the graph of $y = f(x)$ corresponds to the point $(p, q)$ on the graph of $y = g(x)$ , what is the value of $p + q$ ?

-15

-9

-5

3

Show answer & explanation

Answer: $-9$

Answer

-9

The vertex of the quadratic function

f(x)

is identified from the symmetric table values as

(0, 2)

. Under the transformation

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

, the input

x - 3

shifts the vertex horizontally to the right by 3 units, so the new x-coordinate is

p = 3

. Evaluating

g(3)

gives

q = -2(f(0) + 4) = -2(2 + 4) = -2(6) = -12

. The sum of these coordinates is

p + q = 3 + (-12) = -9

.

Step-by-Step Solution

1

Identify the vertex of the function

f(x)

from the table.

Vertex of

f(x)

is

(0, 2)

.

Since

f

is a quadratic function and the table shows symmetry about

x = 0

(with

f(-2) = f(2) = 6

and

f(-4) = f(4) = 18

), the vertex must be at the point where

x = 0

, which gives

f(0) = 2

.

2

Determine the x-coordinate

p

of the corresponding point on the graph of

g(x)

.

p = 3

.

The function

g

is defined as

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

. The expression

f(x - 3)

indicates a horizontal translation of the graph of

f

to the right by 3 units. Therefore, the x-coordinate of the vertex shifts from

0

to

0 + 3 = 3

.

3

Determine the y-coordinate

q

of the corresponding point on the graph of

g(x)

by evaluating

g(3)

.

q = -12

.

Substitute

x = 3

into the definition of

g(x)

:

g(3) = -2(f(3 - 3) + 4) = -2(f(0) + 4)

. Since

f(0) = 2

, this simplifies to

-2(2 + 4) = -2(6) = -12

.

4

Calculate the value of

p + q

.

p + q = -9

.

Adding the coordinates

p = 3

and

q = -12

yields

3 + (-12) = -9

.

Key Concept

Function Notation and Transformations

Question 331Question →

The table below shows some values of the exponential function $f$ , where $f(t) = p \cdot q^t$ for constants $p$ and $q$ .

$t$	$f(t)$
$0$	$80$
$2$	$180$
$4$	$405$

What is the value of $q$ ?

Show answer & explanation

Answer: 1.5

Answer

1.5

The correct answer is 1.5. Since the value of the function at

t = 0

is

80

, the initial value coefficient is

80

. At

t = 2

, the value is

180

, which gives the equation

80 \cdot q^2 = 180

. Solving for

q^2

yields

q^2 = 2.25

, and taking the positive square root gives

q = 1.5

.

Step-by-Step Solution

1

Set up the general exponential equation using the initial value

f(0) = p \cdot q^0 = 80 \implies p = 80

The initial value at

t = 0

directly gives the coefficient

p

because

q^0 = 1

.

2

Substitute another point from the table to solve for the base

q

f(2) = 80 \cdot q^2 = 180 \implies q^2 = 2.25

Using the point

(2, 180)

allows us to write an equation with one variable,

q

.

3

Solve for

q

by taking the square root

q = 1.5

Since the base of an exponential function must be positive, we take the positive square root of 2.25.

Key Concept

Determining the base of an exponential function from a table of values

Question 332Question →

In the quadratic equation $x^2 + kx + 36 = 0$ , $k$ is a constant. If the equation has two distinct real solutions, which of the following could be the value of $k$ ?

-12

-15

-10

8

Show answer & explanation

Answer: $-15$

Answer

The value

-15

is a possible value for

k

because it is the only option that satisfies the inequality

k < -12

or

k > 12

, which guarantees a positive discriminant and thus two distinct real solutions.

A quadratic equation has two distinct real solutions when its discriminant is positive. For

x^2 + kx + 36 = 0

, the discriminant is

k^2 - 4(1)(36) = k^2 - 144

. The inequality

k^2 - 144 > 0

simplifies to

k^2 > 144

, which means

k > 12

or

k < -12

. Among the choices, the value

-15

is the only one that satisfies this condition.

Step-by-Step Solution

1

Identify the condition for a quadratic equation to have two distinct real solutions.

The discriminant,

b^2 - 4ac

, must be strictly greater than

0

.

A positive discriminant ensures the quadratic formula yields two distinct real solutions.

2

Substitute the coefficients of the given equation

x^2 + kx + 36 = 0

into the discriminant formula.

Here,

a = 1

,

b = k

, and

c = 36

. The discriminant is

k^2 - 4(1)(36) = k^2 - 144

.

This expresses the discriminant in terms of the constant

k

.

3

Solve the inequality

k^2 - 144 > 0

for

k

.

k^2 > 144 \Rightarrow |k| > 12

, which means

k > 12

or

k < -12

.

This gives the range of values for

k

that produce two distinct real solutions.

4

Compare the given choices to the solution range.

Only

-15

satisfies the condition because

-15 < -12

.

This determines which of the options is a valid value for

k

.

Key Concept

Determining the number of real solutions of a quadratic equation using the discriminant.

Alternative Method

Instead of solving the inequality algebraically, substitute each answer choice for

k

into the equation and compute the discriminant

b^2 - 4ac

directly to identify which option yields a positive value.

Estimated Time:1m 30s

Question 333Question →

The graph of the quadratic function $f$ in the $xy$ -plane has its vertex at $(3, -4)$ and passes through the point $(1, 8)$ . The function $g$ is defined by $g(x) = f(x - h) + k$ , where $h$ and $k$ are constants. If the graph of $g$ has its vertex at $(0, 0)$ , what is the value of $g(4)$ ?

16

48

12

24

Show answer & explanation

Answer: 48

Answer

48

The quadratic function

f(x)

can be written in vertex form as

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

. Substituting the point

(1, 8)

gives

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

, which simplifies to

12 = 4a

, so

a = 3

. This means

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

. The function

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

has its vertex at

(0, 0)

, which is a translation of the vertex of

f(x)

from

(3, -4)

to

(0, 0)

. This shifts the function to

g(x) = 3x^2

. Evaluating

g(4)

yields

3(4)^2 = 3(16) = 48

.

Step-by-Step Solution

1

Write the vertex form of the quadratic function

f(x)

.

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

The vertex of the parabola is given as

(3, -4)

, so we can substitute these coordinates into the vertex form

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

.

2

Substitute the point

(1, 8)

into the equation to solve for the constant

a

.

8 = a(1 - 3)^2 - 4 \implies 12 = 4a \implies a = 3

Since the graph passes through

(1, 8)

, these coordinates must satisfy the equation of the function.

3

Determine the equation of the transformed function

g(x)

.

g(x) = 3x^2

The vertex of

f(x)

is at

(3, -4)

. Shifting this vertex to

(0, 0)

removes the horizontal and vertical offsets, simplifying the function to

g(x) = ax^2

with the same leading coefficient

a = 3

.

4

Evaluate

g(4)

.

g(4) = 3(4)^2 = 3(16) = 48

Substitute

x = 4

into the equation for

g(x)

to find the required value.

Key Concept

Quadratic Functions and Graphs

Question 334Question →

If $3^{2x - 1} = 81 \cdot 9^{y + 2}$ , which of the following equations correctly expresses $x$ in terms of $y$ ?

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

x = y + \frac{7}{2}

x = y + \frac{9}{2}

x = 4y + \frac{17}{2}

Show answer & explanation

Answer: $x = y + \frac{9}{2}$

Answer

x = y + \frac{9}{2}

The correct equation is found by expressing

81

as

3^4

and

9^{y+2}

as

3^{2y+4}

. Applying the product rule for exponents, the right side becomes

3^{2y+8}

. Since the bases are the same, equating the exponents gives

2x - 1 = 2y + 8

. Solving for

x

yields

x = y + \frac{9}{2}

.

Step-by-Step Solution

1

Rewrite all parts of the equation using a common base of

3

.

Since

81 = 3^4

and

9 = 3^2

, the term

9^{y+2}

becomes

(3^2)^{y+2} = 3^{2y+4}

. The equation can be rewritten as

3^{2x - 1} = 3^4 \cdot 3^{2y + 4}

.

Expressing all exponential terms with the same base allows the use of exponent rules to simplify the equation.

2

Simplify the product on the right side of the equation using the product rule for exponents,

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

.

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

, which simplifies to

3^{2x - 1} = 3^{2y + 8}

.

Adding the exponents of terms with a common base simplifies the right side into a single exponential expression.

3

Set the exponents equal to each other.

2x - 1 = 2y + 8

If two exponential expressions with the same positive base (other than

1

) are equal, their exponents must be equal.

4

Solve for

x

in terms of

y

.

Add

1

to both sides to get

2x = 2y + 9

, then divide by

2

to obtain

x = y + \frac{9}{2}

.

This isolates the variable

x

to express it as a function of

y

.

Key Concept

Solving exponential equations by expressing terms with a common base and applying exponent laws.

Question 335Question →

A parabola in the $xy$ -plane has vertex $(3, 18)$ and passes through the origin. If the equation of the parabola is written in the form $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, what is the value of $a + b$ ?

Show answer & explanation

Answer: 10

Answer

10

The vertex form of a quadratic function with vertex

(h, k)

is

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

. Substituting the given vertex

(3, 18)

yields the equation

y = a(x - 3)^2 + 18

. Since the parabola passes through the origin, we can substitute the point

(0, 0)

into the equation to find the value of

a

:

0 = a(0 - 3)^2 + 18 \implies 9a = -18 \implies a = -2

. Substituting

a = -2

back into the vertex form and expanding gives

y = -2(x - 3)^2 + 18 = -2(x^2 - 6x + 9) + 18 = -2x^2 + 12x

. Comparing this to the standard form

y = ax^2 + bx + c

, we identify

a = -2

and

b = 12

. The sum of these constants is

a + b = -2 + 12 = 10

.

Step-by-Step Solution

1

Write the equation of the parabola in vertex form.

y = a(x - 3)^2 + 18

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex.

2

Substitute the coordinates of the origin

(0, 0)

to solve for

a

.

a = -2

The parabola passes through the origin

(0, 0)

, so these coordinates must satisfy the equation.

3

Expand the vertex form equation into standard form

y = ax^2 + bx + c

.

y = -2x^2 + 12x

Expanding the equation allows us to identify the coefficients

a

and

b

directly.

4

Identify the values of

a

and

b

, and calculate

a + b

.

a + b = 10

Comparing

y = -2x^2 + 12x

to

y = ax^2 + bx + c

gives

a = -2

and

b = 12

, so

a + b = -2 + 12 = 10

.

Key Concept

Vertex form of quadratic functions and conversion to standard form

Question 336Question →

In the $xy$ -plane, the graph of the quadratic function $f$ has its vertex at the point $(4, -3)$ . The function $g$ is defined by $g(x) = f(x + 2) + 5$ . Which of the following ordered pairs represents the vertex of the graph of $g$ ?

(6, 2)

(2, -8)

(2, 2)

(6, -8)

Show answer & explanation

Answer: $(2, 2)$

Answer

(2, 2)

The vertex of

f(x)

is given as

(4, -3)

. The translation

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

shifts the graph

2

units to the left and

5

units upward. Applying these transformations to the vertex of

f(x)

yields the new vertex

(4 - 2, -3 + 5) = (2, 2)

.

Step-by-Step Solution

1

Identify the vertex of the function

f(x)

The vertex of

f(x)

is

(4, -3)

This is given in the problem and serves as the starting point for the transformations.

2

Determine the effect of the horizontal translation

f(x + 2)

The vertex shifts

2

units to the left, changing the

x

-coordinate from

4

to

4 - 2 = 2

For any function

f(x)

, the graph of

f(x + c)

is shifted

c

units to the left when

c > 0

.

3

Determine the effect of the vertical translation

+ 5

The vertex shifts

5

units upward, changing the

y

-coordinate from

-3

to

-3 + 5 = 2

Adding a constant to a function shifts its graph vertically upward by that constant's value.

4

Combine the translated coordinates to find the vertex of

g(x)

The vertex of the graph of

g(x)

is

(2, 2)

The horizontal shift results in an

x

-coordinate of

2

, and the vertical shift results in a

y

-coordinate of

2

.

Key Concept

Quadratic Transformations and Vertex Shifts

Question 337Question →

The function $f$ is defined by $f(x) = 3^x - 4$ . In the $xy$ -plane, the graph of the function $g$ is obtained by first reflecting the graph of $f$ across the $x$ -axis, then translating the graph vertically up by 10 units, and finally translating the graph horizontally to the right by 2 units. If $g(c) = 5$ , what is the value of $c$ ?

Show answer & explanation

Answer: 4

Answer

The value of c is 4.

Reflecting the function

f(x) = 3^x - 4

across the

x

-axis changes its sign to

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

. Translating this graph vertically up by 10 units adds 10 to the function, yielding

-3^x + 14

. Finally, translating horizontally to the right by 2 units replaces

x

with

x - 2

, producing the function

g(x) = -3^{x-2} + 14

. Setting

g(c) = 5

gives the equation

-3^{c-2} + 14 = 5

. Subtracting 14 from both sides results in

-3^{c-2} = -9

, which simplifies to

3^{c-2} = 9

. Since

9 = 3^2

, the exponent

c-2

must equal 2, which gives

c = 4

.

Step-by-Step Solution

1

Reflect the function

f(x) = 3^x - 4

across the

x

-axis

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

A reflection across the

x

-axis replaces

y

with

-y

, meaning the entire function is multiplied by

-1

.

2

Translate the reflected function vertically up by 10 units

-3^x + 4 + 10 = -3^x + 14

A vertical translation upward by

k

units adds

k

directly to the function expression.

3

Translate the function horizontally to the right by 2 units

g(x) = -3^{x-2} + 14

A horizontal translation to the right by

h

units replaces

x

with

x-h

in the function expression.

4

Set

g(c) = 5

and solve the exponential equation for

c

-3^{c-2} + 14 = 5 \implies 3^{c-2} = 9 \implies c - 2 = 2 \implies c = 4

Substitute

c

into

g(x)

, set the output to 5, isolate the exponential term, and equate exponents to find

c

.

Key Concept

Applying sequential function transformations (reflections, vertical translations, horizontal translations) algebraically and solving exponential equations.

Question 338Question →

In the quadratic equation $3x^2 - 15x + c = 0$ , $c$ is a constant. If one of the solutions to the equation is $2$ , what is the value of the other solution?

3

-3

-7

5

Show answer & explanation

Answer: 3

Answer

The other solution to the equation is

3

.

Substituting

x = 2

into the quadratic equation gives

3(2)^2 - 15(2) + c = 0

, which simplifies to

12 - 30 + c = 0

, or

c = 18

. Substituting

c = 18

back into the original equation yields

3x^2 - 15x + 18 = 0

. Dividing the entire equation by

3

results in

x^2 - 5x + 6 = 0

. Factoring this equation gives

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

, which means the solutions are

x = 2

and

x = 3

. Therefore, the other solution is

3

. Alternatively, the sum of the roots of a quadratic equation in the form

ax^2 + bx + c = 0

is given by

-b/a

. For this equation, the sum of the roots is

-(-15)/3 = 5

. Since one root is

2

, the other root must be

5 - 2 = 3

.

Step-by-Step Solution

1

Substitute the given solution

x = 2

into the quadratic equation to find the value of the constant

c

.

3(2)^2 - 15(2) + c = 0 \implies 12 - 30 + c = 0 \implies c = 18

Since

2

is a solution, it must satisfy the equation when substituted for

x

.

2

Substitute

c = 18

back into the original quadratic equation and simplify by dividing both sides by

3

.

3x^2 - 15x + 18 = 0 \implies x^2 - 5x + 6 = 0

Dividing the equation by the greatest common factor simplifies the expression and makes it easier to factor.

3

Factor the simplified quadratic equation to find both solutions.

(x - 2)(x - 3) = 0 \implies x = 2

or

x = 3

The solutions to the factored equation are the values of

x

that make each factor equal to zero.

Key Concept

Solving quadratic equations by substitution and factoring

Estimated Time:1m 30s

Question 339Question →

In the equation $x(x - 8) = k$ , $k$ is a constant. If the product of the two real solutions to the equation is $-20$ , what is the value of the larger solution?

Show answer & explanation

Answer: 10

Answer

The larger solution to the equation is 10.

To find the larger solution, the equation is first rewritten in standard form as

x^2 - 8x - k = 0

. The product of the roots of a quadratic equation in the form

ax^2 + bx + c = 0

is

\frac{c}{a}

. Here,

a = 1

and

c = -k

, so the product of the roots is

-k

. Given that the product of the roots is

-20

, we can set up the equation

-k = -20

, which gives

k = 20

. Substituting

k = 20

back into the equation yields

x^2 - 8x - 20 = 0

. Factoring the quadratic expression gives

(x - 10)(x + 2) = 0

, which has the solutions

x = 10

and

x = -2

. The larger of these two solutions is 10.

Step-by-Step Solution

1

Rewrite the given equation

x(x - 8) = k

in standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 8x - k = 0

To apply quadratic properties, the equation must be in standard form.

2

Use the product of roots formula to find the value of the constant

k

.

-k = -20

, which simplifies to

k = 20

The product of the roots of a quadratic equation in standard form is the constant term divided by the leading coefficient.

3

Substitute the value of

k

back into the quadratic equation and factor it to find the two solutions.

(x - 10)(x + 2) = 0

, so

x = 10

or

x = -2

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

4

Compare the two solutions to identify the larger value.

10

Comparing 10 and -2, 10 is the greater value.

Key Concept

Using the relationship between coefficients and the product of roots to solve a quadratic equation.

Question 340Question →

The graph of $y = f(x)$ has a relative minimum at the point $(2, -4)$ in the $xy$ -plane. If the function $g$ is defined by $g(x) = 3 - f(2x - 6)$ , what are the coordinates of the corresponding relative maximum on the graph of $y = g(x)$ ?

(4, 7)

(-1, 7)

(4, -1)

(6, 7)

Show answer & explanation

Answer: $(4, 7)$

Answer

(4, 7)

To find the coordinates of the relative maximum on the graph of

y = g(x)

that corresponds to the relative minimum of

y = f(x)

at

(2, -4)

, we first find the new

x

-coordinate. The input to the function

f

must be

2

, so we set the argument

2x - 6 = 2

, which solves to

x = 4

. Next, we find the new

y

-coordinate by evaluating

g(4) = 3 - f(2)

. Substituting

f(2) = -4

, we get

g(4) = 3 - (-4) = 7

. Thus, the coordinates of the corresponding relative maximum are

(4, 7)

.

Step-by-Step Solution

1

Set the argument of the function

f

in

g(x)

equal to the

x

-coordinate of the known point on the graph of

f

.

2x - 6 = 2

The relative minimum of

f

occurs when its input is 2.

2

Solve the equation for

x

to find the transformed

x

-coordinate.

2x = 8 \implies x = 4

Isolating the variable

x

by adding 6 to both sides and then dividing by 2.

3

Substitute

x = 4

into the definition of

g(x)

to calculate the transformed

y

-coordinate.

g(4) = 3 - f(2(4) - 6) = 3 - f(2)

This evaluates the vertical transformation of the function at the corresponding

x

-value.

4

Substitute the value

f(2) = -4

and simplify.

g(4) = 3 - (-4) = 3 + 4 = 7

Since the minimum of

f

is at

(2, -4)

, we know

f(2) = -4

. The negation in front of

f

reflects the minimum to a maximum.

Key Concept

Function Notation and Transformations

Estimated Time:2m 0s

Advanced Math

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