1Using Symmetry of a Parabola
The graph of the quadratic function y = f(x) in the xy-plane intersects the x-axis when x = 39 and when x = p, where p is a constant. The maximum value of y = f(x) occurs at the point (14, m), where m is a constant. What is the value of p?
Answer Explanation
A parabola's two x-intercepts are always symmetric about the x-coordinate of its vertex. Since the vertex is at x = 14, that value must be exactly halfway between the two roots: 14 = (39 + p)/2.
Solve for p: 28 = 39 + p, so p = −11.
2Finding a Y-Intercept After a Shift
g(x) = 2(16x − 17). What is the y-intercept of the graph of y = g(x) − 3 in the xy-plane?
Answer Explanation
Find g(0) first: g(0) = 2(16(0) − 17) = 2(−17) = −34.
Subtract 3 to account for the shift: −34 − 3 = −37. The correct answer is A.
3Solving an Absolute Value Function Equation
f(x) = |71 − 2x|. The function f is defined by the given equation. For which of the following values of k does f(k) = 3k?
- A) 71/5
- B) 71/2
- C) 213/5
- D) 71
Answer Explanation
Split into two cases. Case 1: 71 − 2k = 3k, so 71 = 5k, giving k = 71/5. Check: 3(71/5) = 213/5, and 71 − 2(71/5) = 213/5 — both sides match and are non-negative, so this works.
Case 2: 71 − 2k = −3k, so k = −71, giving 3k = −213, which is negative — impossible since an absolute value can never equal a negative number. Only the first case is valid. The correct answer is A.
4Solving a Function Equation
The function f is defined by f(x) = 6(2x + 4). For what value of x does f(x) = 48?
Answer Explanation
Set the function equal to 48: 6(2x + 4) = 48. Divide both sides by 6: 2x + 4 = 8.
Subtract 4: 2x = 4, so x = 2. The correct answer is A.
5Finding a Y-Intercept from a Shifted Table
The table shows three values of x and their corresponding values of y, where y = f(x) + 6 and f is a quadratic function: (20, −20), (24, 12), (28, −20). What is the y-coordinate of the y-intercept of the graph of y = f(x) in the xy-plane?
Answer Explanation
First undo the shift to get f-values: f(20) = −20−6 = −26, f(24) = 12−6 = 6, f(28) = −20−6 = −26. Since f(20) = f(28), the vertex sits exactly halfway between them, at x = 24 — matching the given point, so the vertex is (24, 6).
Write f(x) = A(x−24)² + 6. Use f(20) = −26: A(16) + 6 = −26, so 16A = −32, giving A = −2. Then f(0) = −2(−24)² + 6 = −2(576) + 6 = −1146.
6Finding a Y-Intercept
The y-intercept of the graph of y = 6x² + 77 is (0, y). What is the value of y?
Answer Explanation
Substitute x = 0 into the equation: y = 6(0)² + 77.
y = 77.
7Counting Distinct X-Intercepts
Exactly how many x-intercepts does the graph of f(x) = x(x − 8)²(x − 13)³ have in the xy-plane, where y = f(x)?
Answer Explanation
Each distinct factor gives one distinct x-intercept location, regardless of its multiplicity (a repeated factor just changes whether the graph crosses or touches the axis there, not how many distinct locations there are).
The three distinct roots are x = 0, x = 8, and x = 13. The correct answer is D.
8Interpreting a Function Value in Context
p(v) = 15v. The function p gives the relationship between the instantaneous power p(v), in watts, and the potential difference v, in volts, for a current of 150 amperes. Which statement is the best interpretation of p(3) = 450 in this context?
- A) For a potential difference of 3 volts, the instantaneous power is 450 watts.
- B) For a current of 3 amperes, the instantaneous power is 450 watts.
- C) For an instantaneous power of 3 watts, the potential difference is 450 volts.
- D) For a current of 150 amperes, the potential difference is 450 volts.
Answer Explanation
In p(v), the input v represents potential difference in volts, and the output represents power in watts. So p(3) = 450 means an input of 3 volts produces an output of 450 watts.
The current (150 amperes) is a fixed constant built into the function, not part of what p(3) = 450 is describing. The correct answer is A.
9Finding the Least Integer for No Real Solutions
y = −0.5 and y = x² + 10x + a. In the given system of equations, a is a positive integer constant. The system has no real solutions. What is the least possible value of a?
Answer Explanation
Substitute y = −0.5 into the second equation: −0.5 = x² + 10x + a, or x² + 10x + (a + 0.5) = 0. For no real solutions, the discriminant must be negative: 10² − 4(1)(a + 0.5) < 0.
Simplify: 100 − 4a − 2 < 0, so 98 < 4a, giving a > 24.5. Since a must be a positive integer, the least possible value is 25.
10Finding the Doubling Time of Exponential Growth
f(x) = 70,000(2)^(t/820). The function gives the number of bacteria in a population t minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?
- A) 2
- B) 820
- C) 1,640
- D) 70,000
Answer Explanation
In an exponential function written as A(2)^(t/k), the population doubles every time t increases by exactly k — that's what the base-2 exponential structure means.
Here, k = 820, so the population doubles every 820 minutes. The correct answer is B.
11Writing a Piecewise-Style Cost Equation
The cost of renting a piece of constructing equipment for up to 5 days is $350 for the first day and $175 for each additional day. Which of the following equations gives the cost y, in dollars, of renting the equipment for x days, where x is a positive integer and x ≤ 5?
- A) y = 175x + 175
- B) y = 350x − 175
- C) y = 175x + 350
- D) y = 350x + 175
Answer Explanation
The first day costs a flat $350. Each additional day beyond the first costs $175, and there are (x − 1) additional days: total = 350 + 175(x − 1).
Distribute: 350 + 175x − 175 = 175x + 175. The correct answer is A.
12Evaluating a Square Root Function
The function g is defined by g(x) = √(6x + 7). What is the value of g(3)?
Answer Explanation
Substitute x = 3 into the function: g(3) = √(6(3) + 7) = √(18 + 7) = √25.
g(3) = 5. The correct answer is C.
13Evaluating a Square Root Function
The function g is defined by g(x) = √(6x + 7). What is the value of g(3)?
Answer Explanation
Substitute x = 3 into the function: g(3) = √(6(3) + 7) = √(18 + 7) = √25.
g(3) = 5. The correct answer is C.
14Evaluating a Linear Function
The function f is defined by f(x) = 25x + 80. What is the value of f(x) when x = 2?
Answer Explanation
Substitute x = 2 into the function: f(2) = 25(2) + 80.
f(2) = 50 + 80 = 130. The correct answer is C.
15Writing and Solving a System from a Word Problem
A proposal for a construction project was included on a city election ballot. A TV news report stated that 4 times as many people voted of the proposal as people who voted against it. An internet article reported that 10,500 more people voted in favor of the proposal than voted against it. Based on these data, how many people voted against the proposal?
Answer Explanation
Let A = votes against. Then votes in favor = 4A. The difference between favor and against is 10,500: 4A − A = 10,500.
Combine like terms: 3A = 10,500, so A = 3500.
16Evaluating a Square Root Function
The function f is defined by f(x) = 8 + √x. What is the value of f(36)?
Answer Explanation
Substitute x = 36 into the function: f(36) = 8 + √36.
f(36) = 8 + 6 = 14. The correct answer is B.
