Final Practice Exam

Find the degree of x(x + 5)²(x − 1)⁴ and describe how its graph behaves at the far left and far right.

Select one.

• Degree 3; it rises on both ends.
• Degree 7; it rises on the left and falls on the right.
• Degree 7; it falls on the left and rises on the right.
• Degree 8; it falls on the left and rises on the right.

Find the value of 27^(2⁄3) by taking the cube root before squaring.

Select one.

• 18
• 3
• 9
• 729

A substance decays according to A = A₀ · e^(kt), with k = −0.08 and t measured in days. About how long is its half-life?

Select one.

• about −8.66 days
• about 12.50 days
• about 0.06 days
• about 8.66 days

Which interval is the domain of g(x) = log₅(x + 4)?

Select one.

• [−4, ∞)
• (4, ∞)
• (−4, ∞)
• (0, ∞)

For x ≥ 0, write x^(1⁄2) · x^(1⁄4) as x raised to one exponent.

Select one.

• x^(3⁄4)
• x^(1⁄8)
• x^(2⁄6)
• x^(2⁄8)

If g(x) = 2 · 3ˣ, what is g(3)?

Select one.

• 216
• 18
• 29
• 54

A ball is tossed upward, and its height in feet t seconds later is s(t) = −16t² + 32t + 48. At what time is the ball highest, and what is its height then?

Select one.

• After 3 seconds, at 0 feet
• After 1 second, at 64 feet
• At the moment of launch, at a height of 48 feet
• After 1 second, at 48 feet

A student tries values of x in 2^(x − 1) = 32. Which value makes the equation true?

Select one.

• x = 4
• x = 16
• x = 6
• x = 33

Multiplying both sides of 1 ⁄ (x − 3) + 1 = 6 ⁄ (x² − 9) by x² − 9 produces x² + x − 12 = 0, whose roots are x = −4 and x = 3. Which statements are true? Select all that apply.

Select every option that applies.

• x = 3 is a solution of the original equation
• the original equation has no solution
• x = −4 is a solution of the original equation
• x = −3 is one of the candidates
• x = 3 is an extraneous solution

What horizontal asymptote does (4x − 7) ⁄ (2x + 1) have?

Select one.

• y = 2
• y = 1⁄2
• y = 0
• There is no horizontal asymptote.

The function g(x) = 5 · 4ˣ is exponential. What is g(0), its starting value?

Select one.

• 4
• 5
• 20
• 1.25

Which description fits the graph of g(x) = 5 · 3ˣ?

Select one.

• Its y-intercept is (0, 3), and it nears the x-axis to the left without meeting it
• Its y-intercept is (0, 5), and it meets the x-axis at some negative input
• Its y-intercept is (0, 15), and it nears the line y = 5 to the left
• Its y-intercept is (0, 5), and it nears the x-axis to the left without meeting it

To what power must 10 be raised to give 10,000? Equivalently, find log₁₀(10,000).

Select one.

• 4
• 40
• 1,000
• 5

A tracer dose of 240 milligrams halves every period, so A(t) = 240 · (1⁄2)ᵗ, where t counts half-lives. How many milligrams are left after 4 half-lives?

Select one.

• 15 milligrams
• 60 milligrams
• 120 milligrams
• 30 milligrams

For positive x and y, expand log_b(x⁴y⁵) completely using the properties of logarithms.

Select one.

• 20 log_b(xy)
• 4 log_b(x) + 5 log_b(y)
• 4 log_b(x) · 5 log_b(y)
• log_b(4x) + log_b(5y)

Since the absolute value of a number is its distance from zero, and no distance is negative, the equation |y| = −8 has no solution.

True or false.

• True
• False

A print shop's costs are C = 72 + 4x dollars for x posters, and its revenue is R = 12x dollars. Setting 12x = 72 + 4x, how many posters must it sell to break even?

Select one.

• 6
• 4.5
• 9
• 18

Before reducing (x² + 7x + 12) ⁄ (x + 4), the numerator has to be factored. Which factorization of it is right?

Select one.

• (x + 2)(x + 6)
• (x + 3)(x + 4)
• (x − 3)(x − 4)
• (x + 1)(x + 12)

Which ordered pair names the vertex of g(x) = (x − 4)² + 7?

Select one.

• (−4, 7)
• (4, −7)
• (4, 7)
• (−4, −7)

What monomial results from multiplying 4x³ by 5x²?

Select one.

• 9x⁵
• 20x⁶
• 9x⁶
• 20x⁵

What number is √(36 + 64) once the sum under the radical is worked out first?

Select one.

• 14
• 10
• 100
• 48

Apply h = −b ⁄ (2a) to locate the vertex of g(x) = x² − 10x + 16.

Select one.

• (−5, 91)
• (5, −9)
• (0, 16)
• (10, 16)

The first step in working 4(y + 3) = 5y − 2 is to distribute the 4 over the parentheses. Which equation results?

Select one.

• 4y + 3 = 5y − 2
• 4y − 12 = 5y − 2
• 4y + 12 = 5y − 2
• y + 12 = 5y − 2

Which factored form and pair of roots go with x² + 2x − 15 = 0?

Select one.

• (x + 5)(x − 3) = 0, so x = −5 or x = 3
• (x − 5)(x + 3) = 0, so x = 5 or x = −3
• (x + 5)(x − 3) = 0, so x = 5 or x = −3
• (x + 15)(x − 1) = 0, so x = −15 or x = 1

Rewrite h(x) = 3(x + 2)² − 5 in standard form ax² + bx + c.

Select one.

• 3x² + 7
• 9x² + 36x + 31
• 3x² + 12x + 7
• 3x² + 6x + 7

Written without parentheses, the expression −3⁴ equals which number?

Select one.

• −81
• 81
• −12
• 12

At x = −3, what does the graph of (x + 3)⁴(x − 2) do?

Select one.

• It crosses the x-axis, because −3 is a zero.
• It has a vertical asymptote at x = −3.
• It touches the x-axis and turns back, because the zero has multiplicity 4.
• It crosses the x-axis and flattens out there, because the zero has multiplicity 5.

A gym charges m(x) = 6x + 15 dollars for x classes. Which statement about m(4) and the graph of m is correct?

Select one.

• m(4) = 24, so the point (4, 24) is on the line
• m(4) = 39, so the point (39, 4) is on the line
• m(4) = 39, so the point (4, 39) is on the line
• m(4) = 360, because 6 · 4 · 15 = 360

Which vertical lines are asymptotes of the graph of R(x) = (x − 3) ⁄ (x² − 25)?

Select one.

• x = 3
• x = −5 and x = 5
• y = 0
• x = −25 and x = 25

By the product rule for exponents, 5² · 5⁶ can be rewritten as which single power?

Select one.

• 5¹²
• 25⁸
• 5⁸
• 5⁴

An account holding 2,000 dollars earns 3% interest compounded continuously, so after 5 years A = 2000 · e^(0.03 · 5). Taking e^0.15 ≈ 1.161834, what is the balance?

Select one.

• 2,300.00 dollars
• 2,323.67 dollars
• 2,318.55 dollars
• 2,060.00 dollars

Where does the graph of g(x) = x² + 4x − 21 cross the y-axis?

Select one.

• (0, −25)
• (0, −21)
• (0, 1)
• (−21, 0)

Find the exponent x that makes 5 · 3ˣ = 405 true, working from the powers of three.

Select one.

• 81
• 4
• 400
• 27

True or false: the point (5, 5) satisfies the system of inequalities y > x and x + y < 12.

True or false.

• True
• False

A square garden x meters on a side gets a path 2 meters wide on every side, making a larger square x + 4 meters on a side. Which polynomial gives the larger square's area, (x + 4)²?

Select one.

• x² + 16
• x² + 8x + 16
• x² + 4x + 16
• x² + 8x + 8

For the line y = 4x − 7, identify the slope and the y-intercept.

Select one.

• Slope −7; y-intercept (0, 4)
• Slope −4; y-intercept (0, −7)
• Slope 4; y-intercept (0, −7)
• Slope 4; y-intercept (−7, 0)

A pottery studio's average cost per mug is u(x) = (8x + 300) ⁄ x dollars when it fires x mugs, with u(20) = 23 and u(300) = 9. What does the horizontal asymptote of u tell the studio?

Select one.

• The average cost per mug climbs toward 300 dollars.
• The average cost per mug drops to 0 dollars once enough mugs are fired.
• The average cost per mug stays at 23 dollars.
• The average cost per mug falls toward 8 dollars but never reaches it.

The rational expression (x² + 2x − 8) ⁄ (x − 2) reduces to x + 4 whenever x ≠ 2. What appears on its graph at x = 2?

Select one.

• a zero at (2, 0)
• a vertical asymptote at x = 2
• the point (2, 6) drawn as a solid dot
• a hole at (2, 6)

A farmer builds a rectangular pen from 60 feet of fencing; for a width of w feet, the pen's area is A(w) = w(30 − w) = −w² + 30w square feet. What width makes the area as large as possible, and how large is it?

Select one.

• A width of 15 feet, for 225 square feet
• A width of 30 feet, for 900 square feet
• A width of 18 feet, for 216 square feet
• A width of 10 feet, for an area of 200 square feet

For p(x) = 2x² + 5x − 4, which expression equals p(−x)?

Select one.

• −2x² − 5x + 4
• 2x² + 5x − 4
• 2x² − 5x − 4
• −2x² + 5x − 4