Pre-Requisite Skills
QUESTION: For f(x) = 2x² − 3x + 1, identify a, b, and c.
ANSWER: a = 2, b = −3, c = 1.
A student solves x² = 36 and gives only x = 6. What is missing?
Answer: x = −6. Both 6 and −6 square to 36.
Find f(2) for f(x) = 2x² − x + 1.
Answer: 2(4) − 2 + 1 = 7.
Solve (x − 2)² = 9.
Answer: x − 2 = ±3; x = −1 or 5.
Simplify i² + 1.
Answer: −1 + 1 = 0.
QUESTION: State the vertex of y = (x − 2)² + 3.
ANSWER: (2,3).
A student calls the vertex of y = (x + 3)² − 2 the point (3,−2). Correct it
Answer: (−3,−2), because x + 3 = 0 when x = −3
Expand (x + 3)².
Answer: x² + 6x + 9.
The discriminant is 0. How many distinct real solutions does the quadratic have?
Answer: One distinct real solution, a repeated root.
Solve 50(x − 2)(x + 1) = 0.
Answer: x = 2 or −1. The factor 50 does not create a root.
QUESTION: Find the y-intercept of f(x) = x² + 2x − 6.
ANSWER: f(0) = −6, so (0,−6).
A student finds f(−2) = −1 for f(x) = x² + 3 by writing −4 + 3. Correct the work.
Answer: (−2)² + 3 = 4 + 3 = 7. The negative input must be squared in parentheses.
Factor 2x² + 7x − 4 completely.
Answer: (2x − 1)(x + 4).
An object has height h(t) = −t² + 6t. It reaches height 5 at t = 1 and t = 5. Which time answers “first reaches 5,” and why?
Answer: t = 1, because it is the earlier nonnegative time.
State the vertex of y = 3(x − 4)² + 7.
Answer: (4,7). The coefficient 3 changes the width, not the vertex.
QUESTION: Use the graph. Give both zeros and both x-intercepts.
ANSWER: Zeros −1 and 3; x-intercepts (−1,0), (3,0).
A student writes √(−9) × √(−16) = 12. Correct and explain.
Answer: (3i)(4i) = 12i² = −12.
Write y = (x − 2)² − 5 in standard form.
Answer: y = x² − 4x − 1.
Solve the system y = x² and y = 9. Give both ordered pairs.
Answer: (−3,9) and (3,9).
A quadratic has discriminant −12. How many real solutions does it have?
Answer: No real solutions; two nonreal complex solutions.
A table has x-values 0, 1, 3 and corresponding f(x)-values 2, 5, 11. Find ARC on [0,3].
Answer: (11 − 2)/(3 − 0) = 3.
A student says (x − 4)(x + 2) = 0 has solutions −4 and 2. Correct both and prove one.
Answer: x = 4 or −2. At x = 4, (4 − 4)(4 + 2) = 0; at x = −2, (−2 − 4)(−2 + 2) = 0.
For f(x) = x² + 1, find ARC on [1,3].
Answer: f(1) = 2, f(3) = 10; ARC = (10 − 2)/(3 − 1) = 4.
Profit is modeled by P = x² + 2x. A target profit is 24. Solving gives x = 4 or −6. If x is a selling price, which is meaningful? Explain.
Answer: $4. A negative selling price is not meaningful in this situation.
Use the quadratic formula to solve 2x² + 2x − 4 = 0. The formula is provided.
Answer: a = 2, b = 2, c = −4; D = 36; x = (−2 ± 6)/4 = 1 or −2.