Advanced Math Practice Questions — SAT — Page 19

For the quadratic function $f$ , the table shows some values of $x$ and their corresponding values of $f(x)$ .

$x$	$f(x)$
$1$	$15$
$3$	$3$
$5$	$15$

What is the value of $f(0)$ ?

Show answer & explanation

Answer: 30

Answer

30

Since the function values at

x = 1

and

x = 5

are both

15

, the axis of symmetry of the quadratic function must be located at the midpoint of these values, which is

x = 3

. The vertex of the function must therefore have an

x

-coordinate of

3

. From the table,

f(3) = 3

, which means the vertex is

(3, 3)

. The vertex form of the quadratic function is

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

. To find the value of

a

, substitute the point

(1, 15)

into the equation:

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

, which simplifies to

15 = 4a + 3

, giving

a = 3

. The equation of the function is

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

. Substituting

x = 0

yields

f(0) = 3(0 - 3)^2 + 3 = 3(9) + 3 = 30

.

Step-by-Step Solution

1

Find the axis of symmetry.

The axis of symmetry is

x = 3

.

Since

f(1) = 15

and

f(5) = 15

, the function values are equal at these points. Because a quadratic function is symmetric, the axis of symmetry is the midpoint of these two

x

-values:

\frac{1 + 5}{2} = 3

.

2

Identify the vertex.

The vertex is

(3, 3)

.

The vertex of a quadratic function lies on its axis of symmetry. Since the axis of symmetry is

x = 3

, the

x

-coordinate of the vertex is

3

. From the table,

f(3) = 3

, so the vertex is

(3, 3)

.

3

Write the function in vertex form and solve for the leading coefficient.

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

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex. Substituting

(3, 3)

, we get

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

. Using the point

(1, 15)

from the table, we substitute

x = 1

and

f(x) = 15

:

15 = a(1 - 3)^2 + 3 \Rightarrow 15 = 4a + 3 \Rightarrow 12 = 4a \Rightarrow a = 3

.

4

Evaluate the function at

x = 0

.

f(0) = 30

Substitute

x = 0

into the function:

f(0) = 3(0 - 3)^2 + 3 = 3(9) + 3 = 27 + 3 = 30

.

Key Concept

Quadratic Functions and Graphs

Question 362Question →

In the equation $x^2 - bx + 16 = 0$ , $b$ is a positive integer. If the equation has no real solutions, what is the greatest possible value of $b$ ?

Show answer & explanation

Answer: 7

Answer

The greatest possible value of

b

is

7

.

For the quadratic equation

x^2 - bx + 16 = 0

to have no real solutions, its discriminant must be less than

0

. The discriminant is

(-b)^2 - 4(1)(16) = b^2 - 64

. Solving the inequality

b^2 - 64 < 0

gives

b^2 < 64

. Since

b

is a positive integer, taking the square root of both sides gives

b < 8

. The positive integers less than

8

are

1, 2, 3, 4, 5, 6,

and

7

. The greatest of these values is

7

.

Step-by-Step Solution

1

Set up the inequality for the discriminant to be less than zero.

(-b)^2 - 4(1)(16) < 0

A quadratic equation of the form

ax^2 + bx + c = 0

has no real solutions if and only if its discriminant,

D = b^2 - 4ac

, is negative.

2

Simplify the quadratic inequality.

b^2 < 64

Squaring

-b

yields

b^2

and calculating

4(1)(16)

yields

64

.

3

Solve for the greatest positive integer value of

b

.

b < 8

, so the greatest positive integer is

7

.

Since

b

is a positive integer, the values satisfying

b^2 < 64

are

1, 2, 3, 4, 5, 6,

and

7

. The largest of these is

7

.

Key Concept

Quadratic Discriminant and Number of Solutions

Question 363Question →

If $4^{a - 1} \cdot 8^{2a} = 16^3$ , what is the value of $a$ ?

\\frac{9}{8}

\\frac{13}{8}

\\frac{7}{4}

\\frac{11}{4}

Show answer & explanation

Answer: $\\frac{7}{4}$

Answer

\\frac{7}{4}

The correct answer is

\frac{7}{4}

. Rewriting each base in the equation

4^{a - 1} \cdot 8^{2a} = 16^3

as a power of 2 gives

(2^2)^{a-1} \cdot (2^3)^{2a} = (2^4)^3

. Applying the power of a power rule results in

2^{2a-2} \cdot 2^{6a} = 2^{12}

. Using the product rule of exponents to combine the left side yields

2^{2a-2+6a} = 2^{8a-2} = 2^{12}

. Setting the exponents equal gives

8a - 2 = 12

, which simplifies to

8a = 14

, or

a = \frac{7}{4}

.

Step-by-Step Solution

1

Rewrite each base in the equation

4^{a - 1} \cdot 8^{2a} = 16^3

as a power of 2.

(2^2)^{a-1} \cdot (2^3)^{2a} = (2^4)^3

To solve an exponential equation with different bases, rewrite the bases so they are identical.

2

Apply the power of a power rule

(x^m)^n = x^{m \cdot n}

to simplify each term.

2^{2a-2} \cdot 2^{6a} = 2^{12}

This simplifies the exponents by multiplying the inner and outer exponents.

3

Apply the product rule

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

to combine the terms on the left side.

2^{8a-2} = 2^{12}

This combines the exponents of the terms with the common base of 2.

4

Set the exponents equal to each other and solve the resulting linear equation for

a

.

8a - 2 = 12 \Rightarrow 8a = 14 \Rightarrow a = \\frac{7}{4}

Since the bases are equal, their exponents must be equal.

Key Concept

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

Alternative Method

Instead of converting to base 2, all terms can be written in base 4:

4^{a - 1} \cdot (4^{1.5})^{2a} = (4^2)^3

, which simplifies to

4^{a - 1 + 3a} = 4^6

, leading to

4a - 1 = 6

and

a = \\frac{7}{4}

.

Estimated Time:1m 30s

Question 364Question →

If $27^{x-1} = \sqrt{3^{5x+1}}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 7

Answer

The correct answer is 7.

The correct answer is 7. By converting the base of 27 to

3^3

and rewriting the radical on the right side as a fractional exponent of

\frac{1}{2}

, the equation becomes

3^{3(x-1)} = 3^{\frac{5x+1}{2}}

. Since the bases are equal, their exponents must be equal:

3x - 3 = \frac{5x+1}{2}

. Multiplying both sides by 2 yields

6x - 6 = 5x + 1

. Subtracting

5x

and adding 6 to both sides isolates

x

, giving the solution

x = 7

.

Step-by-Step Solution

1

Express both sides of the equation with a common base of 3.

(3^3)^{x-1} = (3^{5x+1})^{\frac{1}{2}}

Expressing terms with a common base allows the exponents to be compared directly.

2

Apply the power of a power exponent rule,

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

, to simplify the exponents on both sides.

3^{3x-3} = 3^{\frac{5x+1}{2}}

Simplifying the expressions makes it possible to set the exponent expressions equal to each other.

3

Equate the exponents since the bases are identical.

3x - 3 = \frac{5x+1}{2}

If two exponential expressions with the same positive base (other than 1) are equal, their exponents must also be equal.

4

Solve the linear equation for

x

by clearing the fraction and isolating the variable.

x = 7

Multiplying both sides by 2 gives

6x - 6 = 5x + 1

. Subtracting

5x

from both sides and adding 6 to both sides isolates

x

, resulting in

x = 7

.

Key Concept

Solving exponential equations by converting to a common base and applying exponent laws.

Question 365Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, has a vertex at $(2, -3)$ and passes through the point $(4, 5)$ . If the graph of a second quadratic function, $g$ , is obtained by translating the graph of $f$ horizontally by $3$ units to the right and vertically by $5$ units up, what is the value of $g(5)$ ?

-8

2

8

74

Show answer & explanation

Answer: 2

Answer

The value of

g(5)

is

2

.

The vertex of the graph of

f

is at

(2, -3)

, which means

f(2) = -3

. The graph of

g

is obtained by translating the graph of

f

by

3

units to the right and

5

units up, so its equation is

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

. To find

g(5)

, we substitute

x = 5

into this relation, which gives

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

. Since

f(2) = -3

, we have

g(5) = -3 + 5 = 2

. Alternatively, translating the vertex of

f

at

(2, -3)

by

3

units to the right and

5

units up gives the vertex of

g

at

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

. Since the vertex of the parabola

g

occurs at

x = 5

, the value of

g(5)

is the

y

-coordinate of the vertex, which is

2

.

Step-by-Step Solution

1

Identify the vertex of the function

f

and write its vertex form equation.

The vertex of

f

is

(2, -3)

, so the vertex form of the function is

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

.

This allows us to find the specific equation of

f(x)

if needed, and also tells us that the value of

f(2)

is

-3

.

2

Set up the equation for the translated function

g(x)

.

A horizontal translation of

3

units to the right and a vertical translation of

5

units up is represented by

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

.

This defines the function

g

in terms of the function

f

using standard translation rules.

3

Evaluate

g(5)

using the relation from step 2.

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

.

This simplifies the calculation by using the value of

f

at the vertex.

4

Substitute the value of

f(2)

to find

g(5)

.

g(5) = -3 + 5 = 2

.

Since the vertex of

f

is at

(2, -3)

, we know

f(2) = -3

, which yields the final result.

Key Concept

Quadratic functions can be analyzed and transformed using their vertex form and function translation rules.

Estimated Time:1m 30s

Question 366Question →

A retail store models its weekly profit, $P(d)$ , in dollars, from selling a certain product at a discount of $d$ dollars using the function $P(d) = -5(d - 8)^2 + 2,500$ , where $0 \leq d \leq 20$ . Which of the following is the best interpretation of the value 8 in this context?

The maximum weekly profit, in dollars, that the store can achieve.

The discount, in dollars, that results in the maximum weekly profit.

The weekly profit, in dollars, when no discount is offered.

The discount, in dollars, that results in the minimum weekly profit.

Show answer & explanation

Answer: The discount, in dollars, that results in the maximum weekly profit.

Answer

The discount, in dollars, that results in the maximum weekly profit.

The quadratic function

P(d) = -5(d - 8)^2 + 2,500

is given in vertex form,

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

, where

(h, k)

is the vertex of the parabola. In this equation, the vertex is

(8, 2500)

. The leading coefficient is

-5

, which is negative, meaning the parabola opens downward and the vertex represents the maximum point of the function. In this context,

d

is the discount in dollars and

P(d)

is the weekly profit. Therefore, the value 8 represents the discount of 8 dollars that results in the maximum weekly profit.

Step-by-Step Solution

1

Identify the form of the quadratic function.

The function

P(d) = -5(d - 8)^2 + 2,500

is in vertex form,

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

, where

(h, k)

represents the vertex of the parabola.

Recognizing the vertex form allows us to directly identify the vertex coordinates without expanding the equation.

2

Determine the vertex and the direction the parabola opens.

The vertex is

(8, 2,500)

. Since the leading coefficient

a = -5

is negative, the parabola opens downward, meaning the vertex represents a maximum point.

The sign of the leading coefficient determines whether the vertex represents a maximum or a minimum.

3

Interpret the coordinates of the vertex in context.

The variable

d

is the discount in dollars, and

P(d)

is the weekly profit in dollars. Thus, at the vertex, the discount is 8 dollars, which corresponds to the maximum profit of 2,500 dollars.

Matching the coordinates of the vertex to the variables in the context gives the correct real-world meaning.

Key Concept

Interpreting the vertex of a quadratic function in vertex form within a real-world context.

Estimated Time:1m 0s

Question 367Question →

For the function $f$ , selected values of $x$ and $f(x)$ are shown in the table below.

$x$	$f(x)$
$-1$	$4$
$1$	$2$
$3$	$-1$
$5$	$6$

The function $g$ is defined by $g(x) = a \cdot f(x - 2) + 5$ , where $a$ is a constant. If $g(5) = 3$ , what is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

2

Evaluating

g(5)

using the formula

g(x) = a \cdot f(x - 2) + 5

requires finding

f(5 - 2) = f(3)

. From the table,

f(3) = -1

. Substituting these values yields

3 = a(-1) + 5

, which simplifies to

-a = -2

, or

a = 2

.

Step-by-Step Solution

1

Express

g(5)

using the given definition of

g(x)

.

g(5) = a \cdot f(3) + 5

By substituting

x = 5

into the definition

g(x) = a \cdot f(x - 2) + 5

, we obtain

g(5) = a \cdot f(5 - 2) + 5 = a \cdot f(3) + 5

.

2

Find the value of

f(3)

from the table.

f(3) = -1

Looking at the row where

x = 3

in the table, the corresponding output

f(x)

is

-1

.

3

Substitute the known values into the equation for

g(5)

and solve for

a

.

a = 2

Substitute

g(5) = 3

and

f(3) = -1

into the equation to get

3 = a(-1) + 5

. Subtracting 5 from both sides gives

-2 = -a

, which simplifies to

a = 2

.

Key Concept

Evaluating a transformed function using a table of values.

Question 368Question →

The graph of the quadratic function $f(x) = x^2 - 6x + c$ , where $c$ is a constant, has its vertex at $(h, k)$ in the $xy$ -plane. If the graph of $f$ is translated $3$ units to the right and $2$ units down, the vertex of the translated graph lies on the line $y = 2x$ . What is the value of $c$ ?

Show answer & explanation

Answer: 23

Answer

23

To find the constant

c

, we first determine the vertex of the function

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

. The x-coordinate of the vertex of a parabola in the form

y = ax^2 + bx + c

is given by

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

. For this function,

x = -\frac{-6}{2(1)} = 3

. Substituting this back into the function gives the y-coordinate of the vertex:

f(3) = (3)^2 - 6(3) + c = c - 9

. Thus, the original vertex is at

(3, c - 9)

. Translating the graph

3

units to the right increases the x-coordinate of the vertex by

3

, making it

3 + 3 = 6

. Translating the graph

2

units down decreases the y-coordinate of the vertex by

2

, making it

(c - 9) - 2 = c - 11

. The problem states that this new vertex

(6, c - 11)

lies on the line

y = 2x

. Substituting these coordinates into the linear equation gives

c - 11 = 2(6)

, which simplifies to

c - 11 = 12

. Adding

11

to both sides gives the value of

c

as

23

.

Step-by-Step Solution

1

Find the vertex

(h, k)

of the original quadratic function

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

.

The vertex is at

(3, c - 9)

.

The x-coordinate of the vertex of a quadratic function

y = ax^2 + bx + c

is given by

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

. For

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

, we have

h = -\frac{-6}{2(1)} = 3

. Substituting

x = 3

into the function gives the y-coordinate:

k = f(3) = 3^2 - 6(3) + c = c - 9

.

2

Determine the coordinates of the vertex after translating the graph

3

units to the right and

2

units down.

The new vertex is at

(6, c - 11)

.

A translation of

3

units to the right increases the x-coordinate of the vertex by

3

, so the new x-coordinate is

3 + 3 = 6

. A translation of

2

units down decreases the y-coordinate of the vertex by

2

, so the new y-coordinate is

(c - 9) - 2 = c - 11

.

3

Set up an equation using the line

y = 2x

and solve for

c

.

c = 23

Since the translated vertex

(6, c - 11)

lies on the line

y = 2x

, substituting

x = 6

and

y = c - 11

into the line's equation must satisfy it:

c - 11 = 2(6) \Rightarrow c - 11 = 12 \Rightarrow c = 23

.

Key Concept

Determining the vertex of a quadratic function and applying translations to its graph.

Estimated Time:1m 30s

Question 369Question →

In the quadratic equation $x^2 - 6x - k = 0$ , $k$ is a positive constant. If the solutions to the equation are $x = 3 \pm \sqrt{17}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

8

By applying the quadratic formula to

x^2 - 6x - k = 0

, we find the solutions are

x = 3 \pm \sqrt{9 + k}

. Equating the expression inside the radical to the given solutions

3 \pm \sqrt{17}

yields

9 + k = 17

. Solving for

k

gives 8.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation.

a = 1

,

b = -6

, and

c = -k

To apply the quadratic formula, we need to know the values of

a

,

b

, and

c

from the standard form

ax^2 + bx + c = 0

.

2

Substitute the coefficients into the quadratic formula.

x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(-k)}}{2(1)}

The quadratic formula

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

gives the solutions directly.

3

Simplify the expression inside the radical and divide by 2.

x = 3 \pm \sqrt{9 + k}

Simplifying the radical expression allows us to easily compare it to the given solutions.

4

Compare the simplified solution to the given solution form.

9 + k = 17

Since the solutions are given as

3 \pm \sqrt{17}

, the expression inside the radical must equal 17.

5

Solve for the constant

k

.

k = 8

Subtracting 9 from both sides of the equation yields the value of the constant.

Key Concept

Solving quadratic equations using the quadratic formula or by completing the square to find the relationship between coefficients and solutions.

Alternative Method

Instead of using the quadratic formula, the equation can be solved by completing the square. Rewrite the equation as

x^2 - 6x = k

. Adding 9 to both sides gives

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

, which can be factored as

(x - 3)^2 = k + 9

. Taking the square root of both sides yields

x = 3 \pm \sqrt{k + 9}

. Comparing this to the given solutions

3 \pm \sqrt{17}

, we get

k + 9 = 17

, so

k = 8

.

Estimated Time:1m 30s

Question 370Question →

The function $f$ is defined by $f(x) = x^2 - 4x + 7$ . The function $g$ is defined by $g(x) = f(x - 3) + 2$ . If the minimum value of $f$ is $v$ , and the minimum value of $g$ occurs at $x = k$ , what is the value of $v + k$ ?

1

2

7

8

Show answer & explanation

Answer: 8

Answer

8

To find the minimum value of

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

, we can complete the square to write it in vertex form:

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

. The vertex of the graph of

f

is

(2, 3)

, meaning the minimum value of

f

is

v = 3

, occurring at

x = 2

. The function

g

is defined as

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

, which represents a shift of the graph of

f

to the right by 3 units and up by 2 units. Since the minimum of

f

occurs at

x = 2

, the minimum of

g

occurs at

x = 2 + 3 = 5

, so

k = 5

. Thus, the value of

v + k

is

3 + 5 = 8

.

Step-by-Step Solution

1

Find the vertex form of the quadratic function

f(x)

to identify its minimum value

v

and the x-coordinate where it occurs.

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

, so the minimum value is

v = 3

, which occurs at

x = 2

.

Completing the square allows us to read the vertex

(h, k)

of the parabola directly, where

h

is the x-coordinate of the vertex and

k

is the minimum value.

2

Determine the x-coordinate

k

where the minimum value of

g(x)

occurs using function transformations.

k = 2 + 3 = 5

The function

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

shifts the graph of

f

to the right by 3 units. Therefore, the minimum point is translated from

x = 2

to

x = 2 + 3 = 5

.

3

Calculate the sum

v + k

.

3 + 5 = 8

Adding the minimum value of

f

(

v = 3

) to the x-coordinate of the minimum of

g

(

k = 5

) gives the final required value.

Key Concept

Identifying the vertex and minimum values of quadratic functions, and applying horizontal and vertical translations to their graphs.

Estimated Time:1m 30s

Question 371Question →

If $(x - 2)^2 - 9 = 0$ and $x < 0$ , what is the value of $x^2 + 4x$ ?

5

-3

-5

45

Show answer & explanation

Answer: -3

Answer

The value of the expression is -3

To find the value of

x^2 + 4x

, we first solve the equation

(x - 2)^2 - 9 = 0

for

x

. Adding 9 to both sides gives

(x - 2)^2 = 9

. Taking the square root of both sides yields

x - 2 = 3

or

x - 2 = -3

. Solving these two linear equations gives

x = 5

or

x = -1

. The problem states that

x < 0

, so we must choose

x = -1

. Substituting

x = -1

into the expression

x^2 + 4x

gives

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

.

Step-by-Step Solution

1

Isolate the squared term in the equation.

(x - 2)^2 = 9

To solve a quadratic equation of the form

(x - h)^2 - k = 0

, we first add

k

to both sides to isolate the squared binomial.

2

Take the square root of both sides of the equation and solve for

x

.

x - 2 = \pm 3

, which gives

x = 5

or

x = -1

.

Taking the square root of both sides introduces both positive and negative roots, allowing us to find all possible values of

x

.

3

Apply the given constraint

x < 0

to determine the correct value of

x

.

x = -1

The problem specifies that

x

must be less than 0, so the positive solution

x = 5

must be rejected.

4

Substitute the value of

x

into the expression

x^2 + 4x

and simplify.

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

We substitute

x = -1

into the expression to find its final numerical value.

Key Concept

Solving quadratic equations by taking square roots and evaluating expressions under constraints

Estimated Time:1m 30s

Question 372Question →

If $(2y - 3)^2 - 2(2y - 3) - 24 = 0$ and $y > 0$ , what is the value of $y$ ?

\frac{9}{2}

-\frac{1}{2}

\frac{7}{2}

6

Show answer & explanation

Answer: $\frac{9}{2}$

Answer

\frac{9}{2}

Substituting

u = 2y - 3

transforms the original equation into the standard quadratic form

u^2 - 2u - 24 = 0

. Factoring this equation yields

(u - 6)(u + 4) = 0

, giving the solutions

u = 6

and

u = -4

. Substituting

2y - 3

back for

u

results in two equations:

2y - 3 = 6

(which simplifies to

y = \frac{9}{2}

) and

2y - 3 = -4

(which simplifies to

y = -\frac{1}{2}

). Since the problem specifies that

y > 0

, the negative value must be discarded, leaving the correct value as

\frac{9}{2}

.

Step-by-Step Solution

1

Use substitution to simplify the equation by letting

u = 2y - 3

.

u^2 - 2u - 24 = 0

This substitution reduces the equation to a standard quadratic form, making it easier to factor.

2

Factor the quadratic equation

u^2 - 2u - 24 = 0

.

(u - 6)(u + 4) = 0

, which gives

u = 6

or

u = -4

.

Factoring helps identify the potential values for the substituted expression

u

.

3

Substitute

2y - 3

back for

u

and solve both resulting linear equations for

y

.

2y - 3 = 6 \implies 2y = 9 \implies y = \frac{9}{2}

and

2y - 3 = -4 \implies 2y = -1 \implies y = -\frac{1}{2}

.

This step converts the solutions for the intermediate variable

u

back into solutions for the original variable

y

.

4

Apply the given constraint

y > 0

to identify the final solution.

y = \frac{9}{2}

because

-\frac{1}{2}

is not greater than

0

.

The question specifies that

y

must be strictly positive, so we must discard any negative values.

Key Concept

Solving quadratic equations using substitution and factoring under constraints

Question 373Question →

(2x - 5)(2x + 7) = -k

In the given equation, $k$ is a constant. If the equation has exactly one real solution, what is the value of $k$ ?

Show answer & explanation

Answer: 36

Answer

36

Expanding the equation

(2x - 5)(2x + 7) = -k

gives

4x^2 + 4x - 35 = -k

. Adding

k

to both sides puts it in standard form

4x^2 + 4x + (k - 35) = 0

. For a quadratic equation to have exactly one real solution, its discriminant must be zero:

b^2 - 4ac = 4^2 - 4(4)(k - 35) = 0

. Simplifying this yields

16 - 16(k - 35) = 0

, which gives

1 - (k - 35) = 0

, so

k - 35 = 1

, and thus

k = 36

.

Step-by-Step Solution

1

Expand the left side of the equation and write it in standard form.

4x^2 + 4x + (k - 35) = 0

To apply the quadratic discriminant, the equation must be in the form

ax^2 + bx + c = 0

.

2

Identify the coefficients and set the discriminant

b^2 - 4ac

equal to 0.

4^2 - 4(4)(k - 35) = 0

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

3

Solve the resulting linear equation for the constant

k

.

k = 36

Isolating

k

gives the value that satisfies the condition of having exactly one real solution.

Key Concept

Quadratic Discriminant

Question 374Question →

The graph of the quadratic function $f$ in the $xy$ -plane has its vertex at $(4, 12)$ . The function is defined by $f(x) = -(x - c)(x - d)$ , where $c$ and $d$ are constants. What is the value of the product $cd$ ?

Show answer & explanation

Answer: 4

Answer

The value of the product

cd

is

4

.

The

x

-coordinate of the vertex of a quadratic function of the form

f(x) = -(x-c)(x-d)

is the average of the

x

-intercepts

c

and

d

. Since the vertex is

(4, 12)

, we have

\frac{c+d}{2} = 4

, which means

c+d = 8

. Substituting the vertex

(4, 12)

into the function gives

12 = -(4-c)(4-d)

, which expands to

12 = -(16 - 4(c+d) + cd)

. Substituting

c+d = 8

gives

12 = -(16 - 32 + cd) = -(-16 + cd) = 16 - cd

. Solving for

cd

yields

cd = 4

.

Step-by-Step Solution

1

Find the sum of the constants

c

and

d

using the

x

-coordinate of the vertex.

c + d = 8

The graph of a quadratic function in the form

f(x) = -(x-c)(x-d)

has a vertical line of symmetry at the

x

-coordinate of its vertex, which is the midpoint of its

x

-intercepts

c

and

d

. Therefore,

\frac{c+d}{2} = 4

, which simplifies to

c + d = 8

.

2

Substitute the vertex coordinates

(4, 12)

into the function definition.

(4-c)(4-d) = -12

Since

(4, 12)

is the vertex, the point lies on the graph of

f

, meaning

f(4) = 12

. Substituting

x = 4

into the function gives

12 = -(4-c)(4-d)

, which simplifies to

(4-c)(4-d) = -12

.

3

Expand the expression

(4-c)(4-d)

and substitute

c+d = 8

to solve for

cd

.

cd = 4

Expanding

(4-c)(4-d) = -12

gives

16 - 4(c+d) + cd = -12

. Substituting

c+d = 8

yields

16 - 32 + cd = -12

, which simplifies to

-16 + cd = -12

. Adding

16

to both sides gives

cd = 4

.

Key Concept

Quadratic Functions and Graphs

Question 375Question →

The graph of a third-degree polynomial function $f$ in the $xy$ -plane has $x$ -intercepts at $(2, 0)$ , $(-1, 0)$ , and $(k, 0)$ . The $y$ -intercept of the graph of $f$ is $(0, 12)$ . If $f(3) = -48$ , what is the value of $k$ ?

-3

-1

3

6

Show answer & explanation

Answer: -3

Answer

-3

The correct answer is

-3

. By writing the cubic polynomial in factored form as

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

, we can use the

y

-intercept

(0, 12)

to determine that

f(0) = 2ak = 12

, which simplifies to

ak = 6

. Next, using the point

(3, -48)

, we find that

f(3) = 4a(3 - k) = -48

, which simplifies to

3a - ak = -12

. Substituting

ak = 6

into this equation gives

3a - 6 = -12

, which simplifies to

3a = -6

, or

a = -2

. Since

ak = 6

and

a = -2

, solving for

k

yields

k = -3

.

Step-by-Step Solution

1

Write the general factored form of the cubic polynomial using its

x

-intercepts.

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

Since the graph of

f

has

x

-intercepts at

(2, 0)

,

(-1, 0)

, and

(k, 0)

, the factors of

f(x)

must be

(x - 2)

,

(x + 1)

, and

(x - k)

, multiplied by a constant vertical stretch factor

a

.

2

Use the

y

-intercept

(0, 12)

to find an equation relating

a

and

k

.

f(0) = a(0 - 2)(0 + 1)(0 - k) = 2ak = 12 \implies ak = 6

Plugging

x = 0

and

f(0) = 12

into the factored form allows us to find the product of

a

and

k

.

3

Use the given function value

f(3) = -48

to set up a second equation.

f(3) = a(3 - 2)(3 + 1)(3 - k) = 4a(3 - k) = -48 \implies a(3 - k) = -12 \implies 3a - ak = -12

Substituting

x = 3

and

f(3) = -48

into the factored form provides a system of equations to solve for the individual constants.

4

Substitute

ak = 6

into the equation from Step 3 to solve for

a

.

3a - 6 = -12 \implies 3a = -6 \implies a = -2

By replacing

ak

with

6

in the equation

3a - ak = -12

, we can isolate and solve for

a

.

5

Solve for

k

using the values of

a

and

ak

.

Since

a = -2

and

ak = 6

, we divide both sides of

-2k = 6

by

-2

to get

k = -3

.

This isolates the unknown value

k

.

Key Concept

Polynomial Factors and Graphs

Question 376Question →

Let the function $f$ be given by $f(x) = 3x - 5$ . If a second function $g$ is defined in terms of $f$ as $g(x) = 2f(x + 1) + 4$ , what is the value of $x$ for which $g(x) = 12$ ?

Show answer & explanation

Answer: 2

Answer

2

Evaluating

g(x) = 12

gives

2f(x + 1) + 4 = 12

. Subtracting 4 from both sides yields

2f(x + 1) = 8

. Dividing by 2 yields

f(x + 1) = 4

. Substituting

x+1

into the definition

f(x) = 3x - 5

gives

3(x + 1) - 5 = 4

. Simplifying the equation results in

3x + 3 - 5 = 4

, which is

3x - 2 = 4

. Solving for

x

gives

3x = 6

, so

x = 2

.

Step-by-Step Solution

1

Set the equation

g(x) = 12

using the definition of

g(x)

2f(x + 1) + 4 = 12

We are given that

g(x) = 12

and want to find the corresponding value of

x

.

2

Isolate the function term

f(x + 1)

f(x + 1) = 4

Subtract 4 from both sides of the equation to get

2f(x + 1) = 8

, then divide by 2.

3

Use the definition of

f(x)

to express

f(x + 1)

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

Substitute

x + 1

in place of

x

in the function

f(x) = 3x - 5

.

4

Set the expression for

f(x + 1)

equal to 4 and solve for

x

x = 2

Solve the linear equation

3x - 2 = 4

by adding 2 to both sides to get

3x = 6

, then dividing by 3.

Key Concept

Applying multiple transformations to a linear function and solving the resulting equation using function notation.

Question 377Question →

If the quadratic equation $x^2 - bx + 16 = 0$ , where $b$ is a positive constant, has two real solutions such that one solution is $4$ times the other, what is the value of $b$ ?

Show answer & explanation

Answer: 10

Answer

The value of

b

is

10

.

By writing the roots as

r

and

4r

, the quadratic equation can be represented as

(x-r)(x-4r) = x^2 - 5rx + 4r^2 = 0

. Comparing this to the given equation

x^2 - bx + 16 = 0

, we establish that

4r^2 = 16

and

b = 5r

. Solving for

r

gives

r^2 = 4

, which means

r = \pm 2

. Since

b

is a positive constant, we select

r = 2

, yielding

b = 5(2) = 10

. Alternatively, using Vieta's formulas, the product of the roots is

r \cdot 4r = 16 \implies 4r^2 = 16 \implies r = \pm 2

, and the sum of the roots is

r + 4r = b \implies 5r = b

. Since

b > 0

, we find

b = 10

.

Step-by-Step Solution

1

Define the roots in terms of a single variable and express the quadratic equation in factored form.

Let the roots be

r

and

4r

. The factored form is

(x - r)(x - 4r) = 0

.

This allows us to relate the given relationship between the roots to the coefficients of the quadratic equation.

2

Expand the factored quadratic expression.

x^2 - 5rx + 4r^2 = 0

This transforms the equation into the standard form

x^2 + Bx + C = 0

so we can match coefficients.

3

Compare the expanded equation with the given equation

x^2 - bx + 16 = 0

.

4r^2 = 16

and

b = 5r

Matching corresponding coefficients allows us to solve for the unknown variables.

4

Solve the constant term equation for

r

.

r = 2

or

r = -2

Finding the value of

r

is necessary to calculate the value of

b

.

5

Calculate the value of

b

using the constraint that

b

is positive.

b = 5(2) = 10

Using the negative root

r = -2

would yield a negative value for

b

(

b = -10

), which violates the constraint that

b

must be a positive constant.

Key Concept

Relating the roots of a quadratic equation to its coefficients

Estimated Time:1m 30s

Question 378Question →

The polynomial function $p$ is defined by $p(x) = x^3 - 4x^2 - kx + 36$ , where $k$ is a constant. In the $xy$ -plane, the graph of $y = p(x)$ has an $x$ -intercept at $(3, 0)$ . What is the value of $k$ ?

Show answer & explanation

Answer: 9

Answer

The correct answer is 9.

An

x

-intercept at

(3, 0)

indicates that when

x = 3

,

p(x) = 0

. Substituting

x = 3

and

p(3) = 0

into the equation

p(x) = x^3 - 4x^2 - kx + 36

results in

3^3 - 4(3)^2 - k(3) + 36 = 0

. Simplifying the terms gives

27 - 36 - 3k + 36 = 0

. The

-36

and

+36

terms cancel, leaving

27 - 3k = 0

. Solving for

k

gives

3k = 27

, which results in

k = 9

.

Step-by-Step Solution

1

Relate the

x

-intercept to the root of the polynomial function.

p(3) = 0

Since the graph of

y = p(x)

has an

x

-intercept at

(3, 0)

, the value of the function at

x = 3

must be

0

.

2

Substitute

x = 3

into the polynomial expression.

3^3 - 4(3)^2 - k(3) + 36 = 0

By setting the expression equal to

0

, we can solve for the unknown constant

k

.

3

Simplify the expression and solve for

k

.

k = 9

27 - 36 - 3k + 36 = 0

simplifies to

27 - 3k = 0

, which yields

3k = 27

and thus

k = 9

.

Key Concept

Relationship between x-intercepts of a graph and the roots of the polynomial function.

Question 379Question →

One of the solutions to the quadratic equation $x^2 + 8x + c = 0$ , where $c$ is a constant, is $x = -4 + \sqrt{5}$ . What is the value of $c$ ?

-11

11

21

59

Show answer & explanation

Answer: 11

Answer

11

The value of 11 is correct because completing the square of the equation

x^2 + 8x + c = 0

yields

(x + 4)^2 = 16 - c

, which gives solutions of the form

x = -4 \pm \sqrt{16 - c}

. Comparing this to the given solution

x = -4 + \sqrt{5}

shows that

16 - c = 5

, which simplifies to

c = 11

. Direct substitution of the solution back into the equation also confirms this result.

Step-by-Step Solution

1

Express the quadratic equation in vertex form by completing the square.

x^2 + 8x + c = (x + 4)^2 - 16 + c = 0

Completing the square allows us to easily solve for the variable in terms of the constant.

2

Isolate the squared term and solve for the variable.

(x + 4)^2 = 16 - c \implies x = -4 \pm \sqrt{16 - c}

Taking the square root of both sides gives the general solutions of the quadratic equation.

3

Equate the general solution to the given solution to find the value of the constant.

Since one solution is

-4 + \sqrt{5}

, we set

16 - c = 5

, which gives

c = 11

.

Matching the terms under the radical determines the value of the constant.

Key Concept

Solving quadratic equations by completing the square or using the quadratic formula

Question 380Question →

The graph of the quadratic function $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, is a parabola in the $xy$ -plane that passes through the points $(-3, 22)$ and $(9, 22)$ . If the minimum value of $f(x)$ is $4$ , what is the value of $f(1)$ ?

Show answer & explanation

Answer: 6

Answer

The correct answer is 6.

The correct answer is 6. The axis of symmetry of the parabola is halfway between the points with equivalent y-values: x = 3. Using the minimum value of 4, the vertex is identified as (3, 4). Writing the equation in vertex form as f(x) = a(x - 3)^2 + 4 and substituting (9, 22) yields a = 0.5. Evaluating the function f(x) = 0.5(x - 3)^2 + 4 at x = 1 yields 6.

Step-by-Step Solution

1

Determine the axis of symmetry of the parabola.

The axis of symmetry is

x = 3

.

Because the parabola passes through the points

(-3, 22)

and

(9, 22)

, which have the same

y

-coordinate, the axis of symmetry must lie halfway between their

x

-coordinates:

x = \frac{-3 + 9}{2} = 3

.

2

Write the quadratic function in vertex form.

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

Since the function has a minimum value of

4

, the vertex of the upward-opening parabola is at

(3, 4)

.

3

Solve for the leading coefficient

a

.

a = 0.5

Substitute the point

(9, 22)

into the vertex form equation:

22 = a(9 - 3)^2 + 4

, which simplifies to

18 = 36a

, so

a = 0.5

.

4

Evaluate

f(1)

.

f(1) = 6

Substitute

x = 1

into the completed function

f(x) = 0.5(x - 3)^2 + 4

to get

f(1) = 0.5(1 - 3)^2 + 4 = 0.5(4) + 4 = 6

.

Key Concept

Using symmetry, vertex form, and given points to determine a quadratic function's equation and evaluate it.

Advanced Math

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer