Math · Unit 14

Functions and Graphs

The coordinate plane, functions, slope and graph shapes

A function is a rule with exactly one output per input, and a graph is that rule made visible. Holding both pictures at once is what the unit is for.

It covers the coordinate plane, slope and linear graphs, quadratics and parabolas, exponential and other shapes, and functions as objects in their own right.

This unit breaks down into 21 short steps and 120 questions, starting at difficulty 3 and building to 4. Below you can see exactly what it covers, how the path is structured, and worked examples with explanations.

Steps: 21
Questions: 120
Difficulty: 3-4

What this unit covers

Slope and Linear Graphs
Parabolas and Quadratic Graphs
The Coordinate Plane
Exponential and Other Graphs
Functions

Where this fits

Needs Algebra. Directly required for Precalculus and Calculus.

Where people slip

Not every relation is a function. If one input can give two outputs - as with a sideways parabola - it fails, and the vertical line test is how you check.

How the unit is structured

Functions and Graphs runs as 21 short steps that unlock in order. 15 are practice rounds and 6 are challenge rounds that pull together everything before them. Questions start at difficulty 3 and climb to 4 as you progress.

Step 1 · easierStep 21 · harder

Challenge rounds

Example questions

30 real questions from this unit, with the answer and the reason behind it, grouped by what they practise. There are 120 in the unit altogether.

Exponential and Other Graphs

Build the sentenceLevel 3
1. Build the true sentence about exponential graphs.
Answer: An exponential graph grows faster and faster
An exponential graph keeps climbing more and more steeply.
Fact or fibLevel 3
2. The graph of y = 2^x passes through the point (0, 1).
Answer: True
Any exponential y = b^x passes through (0, 1) because b^0 = 1.
Fill the blankLevel 3
3. The graph of y = |x| (absolute value) has a ____ shape.
- Vcorrect
- U
- straight line
- circular
The absolute value graph forms a V with its point at the origin.
Match the pairsLevel 3
4. Match each function to the shape of its graph.
Answer: y = 2^x = Exponential curve; y = |x| = V shape; y = x^2 = Parabola
Each of these functions has its own recognizable graph shape.
Multiple choiceLevel 3
5. The graph of y = 2^x is an example of what kind of growth?
- Exponential growthcorrect
- Linear growth
- No growth
- Negative growth
As x increases, y = 2^x keeps doubling, which is exponential growth.
Odd one outLevel 3
6. Which of these is NOT an exponential function?
- y = x^2correct
- y = 2^x
- y = 3^x
- y = (1/2)^x
y = x^2 is a quadratic (the variable is the base); the others have the variable in the exponent.

Functions

Build the sentenceLevel 3
7. Build the true sentence about functions.
Answer: A function gives each input one output
Each input of a function has exactly one output.
Choose all that applyLevel 3
8. For f(x) = 2x, which statements are true? Select all that apply.
- f(0) = 0correct
- f(3) = 6correct
- f(-1) = -2correct
- f(5) = 12
f(0) = 0, f(3) = 6, and f(-1) = -2 are all correct; f(5) = 10, not 12.
Fact or fibLevel 3
9. The notation f(x) is read as 'f of x'.
Answer: True
f(x) means the output of function f at input x, read 'f of x'.
Fill the blankLevel 3
10. A relation where each input has exactly one output is called a ____.
- functioncorrect
- slope
- graph
- point
A function assigns each input exactly one output.
Guess the numberLevel 3
11. If f(x) = 2x + 3, what is f(5)?
Answer: 13
f(5) = 2 x 5 + 3 = 13.
Match the pairsLevel 3
12. For f(x) = x + 10, match each input to its output.
Answer: f(1) = 11; f(5) = 15; f(0) = 10
Add 10 to each input to find the output.

Parabolas and Quadratic Graphs

Fill the blankLevel 3
13. The lowest or highest point of a parabola is called the ____.
- vertexcorrect
- slope
- origin
- root
The turning point of a parabola is its vertex.
Multiple choiceLevel 3
14. What is the vertex of the parabola y = x^2?
- (0, 0)
- (1, 1)
- (0, 1)
- (1, 0)
The lowest point of y = x^2 sits at the origin, (0, 0).
Odd one outLevel 3
15. Three of these parabolas open upward. Which one opens downward?
- y = -2x^2correct
- y = x^2
- y = 3x^2
- y = 0.5x^2
Only y = -2x^2 has a negative leading coefficient, so it opens down.
Build the sentenceLevel 4
16. Build the true sentence about a function's zeros.
Answer: The zeros of a function are its x intercepts
The zeros of a function are where its graph crosses the x-axis.
Choose all that applyLevel 4
17. Which statements about y = x^2 - 4x + 3 are true? Select all that apply.
- x = 1 is a rootcorrect
- x = 3 is a rootcorrect
- The y-intercept is 3correct
- It opens downward
It factors as (x - 1)(x - 3), so x = 1 and x = 3 are roots and its y-intercept is 3; it opens up, not down.
Match the pairsLevel 4
18. Match each parabola to its vertex.
Answer: y = (x - 1)^2 = (1, 0); y = x^2 + 5 = (0, 5); y = (x + 2)^2 = (-2, 0)
In vertex form y = (x - h)^2 + k the vertex is (h, k).

Slope and Linear Graphs

Build the sentenceLevel 3
19. Build the true sentence about slope.
Answer: Slope is equal to rise over run
Slope measures steepness as rise divided by run.
Choose all that applyLevel 3
20. Which of these lines have a positive slope? Select all that apply.
- y = 2x + 1correct
- y = 5x - 3correct
- y = -x + 4
- y = -3x
A positive coefficient of x means a positive slope, so y = 2x + 1 and y = 5x - 3 rise.
Fact or fibLevel 3
21. A vertical line has an undefined slope.
Answer: True
A vertical line has no horizontal run, so rise/run divides by zero and the slope is undefined.
Fill the blankLevel 3
22. In the form y = mx + b, the letter m stands for the ____.
- slopecorrect
- y-intercept
- x-value
- origin
In y = mx + b, m is the slope of the line.
Multiple choiceLevel 3
23. In the line y = 3x + 5, what is the y-intercept?
- 5correct
- 3
- -5
- 8
In y = mx + b the value b is the y-intercept, so it is 5.
Odd one outLevel 3
24. Three of these are straight lines. Which is the odd one out?
- y = x^2correct
- y = 2x + 1
- y = -x + 3
- y = 5
y = x^2 is a curved parabola; the others are straight lines.

The Coordinate Plane

Build the sentenceLevel 3
25. Build the true sentence about points on the plane.
Answer: Each point on the plane has an x and a y coordinate
Every point is located by its x-coordinate and its y-coordinate.
Choose all that applyLevel 3
26. Which of these points lie in Quadrant I? Select all that apply.
- (2, 3)
- (5, 1)
- (-2, 3)
- (4, -1)
Quadrant I has both coordinates positive, so (2, 3) and (5, 1) qualify.
Fact or fibLevel 3
27. In the ordered pair (x, y), the first number tells you how far to move left or right.
Answer: True
The x-coordinate is horizontal (left or right); y is vertical.
Fill the blankLevel 3
28. The horizontal number line on a coordinate plane is called the ____-axis.
- xcorrect
- y
- z
- origin
The horizontal axis is the x-axis.
Match the pairsLevel 3
29. Match each point to its quadrant.
Answer: (2, 3) = Quadrant I; (-2, 3) = Quadrant II; (-2, -3) = Quadrant III
The signs of x and y decide the quadrant.
Guess the numberLevel 4
30. What is the distance between (0, 0) and (6, 8)?
Answer: 10
Distance = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10.

Where these questions come from. Each unit starts as a plan of the concepts it should cover and the difficulty it should span. Questions are written against that plan with AI assistance, then checked by a validator that rejects anything without a single defensible answer, an explanation, or plausible wrong options. How we write questions sets out the whole process, and corrections are fixed in the bank and reach the site and the app the same day.

How you practise

This unit mixes 14 different question formats, so you are recalling and applying rather than recognising the same layout every time.

Build the sentence
Choose all that apply
Fact or fib
Fill the blank
Guess the number
Match the pairs
Multiple choice
Odd one out
Picture question
Put in order
Sort into groups
Spell it
True or false
Type the answer

Practise Functions and Graphs

120 questions across 21 steps. Start with step one and crawl at your own pace.

Play this unit

Read about Functions and Graphs

Explainers from our blog on what this unit covers. Each one ends with real questions from the bank.

How to Find Slope: A Step-by-Step Math GuideHow to find slope becomes simple with rise over run. Learn the slope formula, positive and negative slopes, graphs, points, and common mistakes.August 14, 2026 · 5 min read

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