Hard Questions - Exam Practice

Q: What does it mean for a condition like f(8)=0 and f(4)<-16 to imply something must be true?

These questions translate the function definition into inequalities or equalities involving the constants. The correct choice is the one that follows logically from both the zero condition and the comparison at the specified input value.

Q: How are function transformations such as g(x)=f(x-2) tested in multiple-choice format?

The questions rewrite the shifted input into the original function expression by substituting x-2 wherever x appears in f. The correct option is the one whose formula matches this substitution and simplification.

Aug 25, 2026

Uploaded by Hazel

Hard Questions presents a set of challenging multiple-choice mathematics problems covering coordinate geometry, linear functions with transformations, radicals and algebraic constraints, polynomial factoring and solution-sum properties, exponential and logarithmic-like base form equivalences, and application word problems. It includes investment interest accumulation with piecewise growth rules, factoring with integer parameter bounds, quadratic function characteristics using vertex and symmetry conditions, and function composition through shifts. The worksheet emphasizes selecting the statement that must be true or the correct equation equivalent to a given description.

LanguageEnglish

CategoryExam

FormatPDF

Pages35

Reading Time1 h 28 min

AccessFree to download

Problem Set

Algebra and Functions

Geometry on the Coordinate Plane

Exponential and Growth Models

Word Problems and Applications

Hard Questions 1. In the xy-plane, line k and the line l are perpendicular and intersect at the point (2 , 8) . If line k is defined by the equation y = mx + b, where m and b are constants and m > 1, which of the following points lies on line l? A) (3 , 8 − ~~1~~m) B) (3 , 8 + ~~1~~m) C) (3 , 8 − m) D) (3 , 8 + m) 2. The function f is defined by f(x) = m√−x − n, where m and n are constants. If f(8) = 0 and f(4) < −16, which of the following must be true? A) m > −8 B) n + m = 0 C) n + m > 0 D) n − m > 0 3. (3x + p)(5x2 − 45)(2x2 − 16x + 6p) = 0 In the given equation, p is a positive constant. The sum of the solutions to the equation is ~~20~~3. What is the value of p? 4. For the function f, for every increase in ~~1~~2 in the value of x, the value of f(x) increases by a factor of c, where c is a constant. Which of the following equivalent forms of the function f displays the value of c as the base or the coefficient? A) f (x) = 38(2)6x B) f (x) = 38(8)2x C) f (x) = 38(64)x D) f (x) = 38(4 , 096) ~~x~~2 5. A square map has a side length of 35 inches, and 1 inch on the map represents an actual distance of 17 miles. A smaller version of the same map is printed as a square with the side length 55% shorter than the side length of the previous map. On the smaller map, which of the following is closest to the actual distance, in miles, represented by 1 inch? A) 7.65 B) 10.97 C) 19.25 D) 37.78 @satashkent 1 6. A certain investment account offers a special interest rate for the first 4 months that the account is open followed by a lower interest rate for the rest of the time the account is open. Akeyla opened one of these accounts with an original balance of the account of $700.00 and did not make any other deposits or withdrawals. 4 months after Akeyla opened the account, the balance had increased by 0 .6% of the original balance. 6 months after Akeyla opened the account, the balance had increased by an additional 0.1% of the balance atthe end of the first 4 months. Every 2 months after the first 6 months, the balance had increased by an additional 0 . 1% of the balance 2 months earlier. Which of the following equations could represent the account balance B (x) in dollars, x months after the account was opened, where x ≥ 4? A) B (x) = 704 .20(1 .001) ~~x ~~−2~~4~~ x−4 B) B (x) = 700 .00(1 .001) 2 C) B (x) = 704 .20(1 .001)2(x−4) D) B (x) = 700 .00(1 .001)2(x−4) 7. 34z14 + bz7 + 70 In the given expression, b is a positive integer. If qz 7 + r is a factor of expression, where q and r are positive integers, what is the highest possible value of b? 8. f (x) = ax2 + 6x + c In the given quadratic function, a and c are constants. The graph of y = f (x) in the xy-plane is a parabola that opens upward and has a vertex at the point (h, k), where h and k are constants. If k > 0 and f (−7) = f(5), which of the following must be true? I. c > 3 II. a > 3 A) I only B) II only C) I and II D) Neither I nor II 9. A group of friends decided to divide the $800 cost of a trip equally among themselves. When two of the friends decided not to go on the trip, the remaining two still divided the $800 cost equally, but each friend’s share of the cost increased by $20 . How many friends were in the group originally? @satashkent 2 10. The function f is defined by f(x) = −ax +b, where a and b are constants. In the xy plane, the graph of y = f(x) − 12 has a y-intercept at 􀀒 0 , − ~~ ~~757􀀓 . The product of a and b is 320 . What is the value of a? 11. 8rx2 − 80sx + 8r = 0 r For the given equation, r and s are constants such that r > 0, s < 0 and > k. What is s the least value of k that ensures that the equation has two distinct real solutions? 12. For the function f, for each increase in the value of x by c, where c is a positive constant, the value of f (x) increases by a factor of 16 . Which of the following equivalent forms of the function f displays ~~1~~c as

FAQs

How do perpendicular intersecting lines determine points on the second line in these questions?

The problems use the fact that slopes of perpendicular lines are negative reciprocals. Once the equation of one line is in slope-intercept form, the point lies on the other line if it satisfies that line’s equation after applying the perpendicular relationship.

What does it mean for a condition like f(8)=0 and f(4)<-16 to imply something must be true?

These questions translate the function definition into inequalities or equalities involving the constants. The correct choice is the one that follows logically from both the zero condition and the comparison at the specified input value.

How are function transformations such as g(x)=f(x-2) tested in multiple-choice format?

The questions rewrite the shifted input into the original function expression by substituting x-2 wherever x appears in f. The correct option is the one whose formula matches this substitution and simplification.

Aug 25, 2026

Uploaded by Hazel