NBT NBT Mathematics: Functions and Graphs 2 — Questions and Answers
Question 1: The graph of y = ax + b passes through (0, 4) and (2, 0). What are the values of a and b?
- a = −2, b = 4 (Correct answer)
- a = 2, b = −4
- a = −4, b = 2
- a = 4, b = −2
Correct answer: a = −2, b = 4
y-intercept gives b = 4; gradient = (0 − 4) ÷ (2 − 0) = −2, so a = −2.
Question 2: Which of the following is the equation of a parabola with vertex at (2, −3) and passing through (0, 1)?
- y = (x − 2)² − 3 (Correct answer)
- y = (x + 2)² − 3
- y = −(x − 2)² − 3
- y = (x − 2)² + 3
Correct answer: y = (x − 2)² − 3
Vertex (2, −3) gives y = a(x − 2)² − 3. Substituting (0, 1): 1 = 4a − 3 → a = 1. So y = (x − 2)² − 3.
Question 3: The function p(x) = 2^x. The graph of q(x) is p(x) shifted 3 units upward. What is q(x)?
- q(x) = 2^x + 3 (Correct answer)
- q(x) = 2^(x+3)
- q(x) = 2^x − 3
- q(x) = 3 · 2^x
Correct answer: q(x) = 2^x + 3
A vertical shift of 3 units upward adds 3 to the function: q(x) = 2^x + 3.
Question 4: The domain of f(x) = √(x − 5) is:
- x ≥ 5 (Correct answer)
- x > 5
- x ≤ 5
- all real numbers
Correct answer: x ≥ 5
The expression under the square root must be ≥ 0: x − 5 ≥ 0 → x ≥ 5.
Question 5: A graph shows f(x) crossing the x-axis at x = −3 and x = 2, with a y-intercept at y = −6. Which function fits?
- f(x) = (x + 3)(x − 2) (Correct answer)
- f(x) = (x − 3)(x + 2)
- f(x) = −(x + 3)(x − 2)
- f(x) = (x + 3)(x + 2)
Correct answer: f(x) = (x + 3)(x − 2)
Roots at −3 and 2 → f(x) = a(x + 3)(x − 2). At x = 0: a(3)(−2) = −6 → −6a = −6 → a = 1.
Question 6: What is the y-intercept of the exponential function f(x) = 5 · (0.4)^x?
- (0, 5) (Correct answer)
- (0, 0.4)
- (0, 2)
- (0, 1)
Correct answer: (0, 5)
f(0) = 5 · (0.4)^0 = 5 · 1 = 5, so y-intercept is (0, 5).
Question 7: For f(x) = x² − 6x + 5, find the vertex by completing the square.
- (3, −4) (Correct answer)
- (3, 4)
- (−3, −4)
- (6, 5)
Correct answer: (3, −4)
f(x) = (x − 3)² − 9 + 5 = (x − 3)² − 4; vertex = (3, −4).
NBT NBT Mathematics: Functions and Graphs 2
The graph of y = ax + b passes through (0, 4) and (2, 0).
What are the values of a and b?