Functions and Graphs Practice Questions
ATAR (Australia) · ATAR Mathematics Methods · 141 free MCQs with instant results and detailed explanations.
141
Total
43
Easy
69
Medium
29
Hard
Start Practicing Functions and Graphs
Take a timed quiz or customize your practice session
Sample Questions from Functions and Graphs
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
What is the range of the function f(x) = x^2 for all real numbers x?
A.
All real numbers
B.
All positive real numbers
C.
Non-negative real numbers
D.
Negative real numbers
Show Answer & Explanation
Correct Answer: C
The function f(x) = x^2 outputs non-negative values for all real inputs, hence its range is non-negative real numbers.
Q2
Easy
If the function g(x) = 3x + 4 is graphed, what is the y-intercept of this linear function?
A.
3
B.
4
C.
7
D.
12
Show Answer & Explanation
Correct Answer: B
The y-intercept of a linear function in the form g(x) = mx + b is the value of b, which is 4 in this case.
Q3
Easy
What type of transformation is applied to the function h(x) = (x - 2)^2 + 3 from the parent function f(x) = x^2?
A.
Vertical shift up
B.
Vertical shift down
C.
Horizontal shift left
D.
Horizontal shift right
Show Answer & Explanation
Correct Answer: A
The function h(x) shifts the parent function f(x) = x^2 vertically upwards by 3 units and horizontally to the right by 2 units.
Q4
Medium
Which transformation maps the graph of y = x^2 to y = (x - 3)^2 + 2?
A.
Translation 3 units right and 2 units up
B.
Reflection across the x-axis
C.
Translation 2 units left and 3 units down
D.
Vertical stretch by a factor of 2
Show Answer & Explanation
Correct Answer: A
The equation y = (x - 3)^2 + 2 indicates a translation of the basic parabola y = x^2. The graph shifts 3 units to the right and 2 units up based on the transformations of the vertex form.
Q5
Medium
The function f(x) = 4 - (x - 2)^2 has its maximum at which point?
A.
(2, 4)
B.
(4, 0)
C.
(0, 4)
D.
(2, 0)
Show Answer & Explanation
Correct Answer: A
The function is a downward-opening parabola since the coefficient of the squared term is negative. The vertex is at (2, 4), which is also the maximum point of the parabola.
Q6
Medium
What is the inverse of the function f(x) = 3x + 5?
A.
f^{-1}(x) = (x - 5)/3
B.
f^{-1}(x) = 3(x - 5)
C.
f^{-1}(x) = (x + 5)/3
D.
f^{-1}(x) = 5/x
Show Answer & Explanation
Correct Answer: A
To find the inverse, we swap x and y in the equation y = 3x + 5, leading to x = 3y + 5. Solving for y gives y = (x - 5)/3, which is the inverse function.
Q7
Medium
What is the transformation of the function f(x) = x^2 when it becomes g(x) = (x - 3)^2 + 2?
A.
Shifted left 3 units and up 2 units.
B.
Shifted right 3 units and up 2 units.
C.
Shifted right 3 units and down 2 units.
D.
Shifted left 3 units and down 2 units.
Show Answer & Explanation
Correct Answer: B
The function g(x) = (x - 3)^2 + 2 is derived from f(x) by shifting it right 3 units and up 2 units due to the modifications in the equation.
Q8
Hard
Given the function f(x) = 2x^3 - 6x^2 + 4x, what is the x-coordinate of the local maximum point?
A.
1
B.
2
C.
0
D.
3
Show Answer & Explanation
Correct Answer: B
To find the local maximum, we take the derivative f'(x) = 6x^2 - 12x + 4 and set it to zero. Solving 6x^2 - 12x + 4 = 0 gives x = 2. This is a local maximum because the second derivative test shows f''(2) < 0.
Q9
Hard
Consider the function g(x) = x/(x^2 - 4). Which of the following statements is true about the vertical asymptotes of g?
A.
At x = -2 and x = 2
B.
At x = 0
C.
At x = 4
D.
At x = 1
Show Answer & Explanation
Correct Answer: A
Vertical asymptotes occur where the denominator is zero and the function is undefined. For g(x), the denominator is x^2 - 4 = 0, which factors to (x - 2)(x + 2) = 0, giving vertical asymptotes at x = -2 and x = 2.
Q10
Hard
Consider the function f(x) = 2x^3 - 3x^2 + 4. What is the value of x for which the function has a local maximum?
A.
1
B.
2
C.
0
D.
4
Show Answer & Explanation
Correct Answer: A
To find local maxima, we need to calculate the derivative f'(x) = 6x^2 - 6 and set it to zero. Solving 6x^2 - 6 = 0 gives x = 1, which corresponds to a local maximum.
Showing 10 of 141 questions. Start a quiz to practice all questions with scoring and timer.
Practice All 141 Questions →More ATAR Mathematics Methods Chapters
Other ATAR (Australia) Subjects
Functions and Graphs — ATAR (Australia) ATAR Mathematics Methods Practice Questions Online
This page contains 141 practice MCQs for the chapter Functions and Graphs in ATAR (Australia) ATAR Mathematics Methods. The questions are organized by difficulty — 43 easy, 69 medium, 29 hard — so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.