Year 11 · Mathematics · Algebra
Functions and their graphs
Use function notation, find the domain and range of common functions, and describe how graphs are moved by simple transformations.
Awaiting educator review. This material was drafted by our content team with AI assistance and is waiting for review by a qualified educator. Please check it suits your child before using it. Created 10 October 2026.
Learning goals
Learners will
- Understand what makes a relation a function.
- Use function notation to evaluate functions.
- Find the domain and range of common functions.
- Describe translations of the graph of y = x^2.
Success looks like
- I can use the vertical line test to decide whether a graph is a function.
- I can evaluate f(a) for a given function and value a.
- I can state the domain and range of linear, quadratic, square root and simple rational functions.
- I can describe how y = (x − h)^2 + k is related to y = x^2.
The big idea
A relation is any set of ordered pairs (x, y). A function is a special relation in which every input x has exactly one output y. For example, y = x^2 is a function because each x gives just one y, even though different xs (such as 2 and −2) can give the same y.
On a graph, use the vertical line test: if any vertical line crosses the graph more than once, the graph is not a function. A circle such as x^2 + y^2 = 9 fails this test, because a vertical line through its middle crosses it twice.
Function notation names a function and its input. If f(x) = 2x^2 − 3, then f(4) means "replace x with 4": f(4) = 2 × 16 − 3 = 29. Use brackets when substituting negative numbers: f(−1) = 2 × (−1)^2 − 3 = −1.
The domain is the set of x-values you are allowed to use. Most polynomial functions accept every real number. Two common restrictions are: you cannot take the square root of a negative number, and you cannot divide by zero. So sqrt(x − 2) needs x ≥ 2, and 1/(x − 3) needs x ≠ 3.
The range is the set of y-values the function actually produces. A sketch is the most reliable way to find it. For y = x^2 + 1, the lowest point is at y = 1 and the graph rises forever, so the range is y ≥ 1.
Graphs can be translated. The graph of y = (x − h)^2 + k is the parabola y = x^2 moved h units right and k units up, so its vertex is at (h, k). Watch the sign inside the bracket: y = (x + 2)^2 moves the graph 2 units to the left, because x + 2 = x − (−2).
Worked examples
Evaluating a function
If f(x) = 2x^2 − 3, find f(4) and f(−1).
- f(4) = 2 × 4^2 − 3 = 2 × 16 − 3 = 32 − 3 = 29.
- f(−1) = 2 × (−1)^2 − 3 = 2 × 1 − 3 = −1.
Answer: f(4) = 29 and f(−1) = −1
Finding a domain
State the domain of f(x) = sqrt(x − 2).
- The expression under a square root cannot be negative.
- So x − 2 ≥ 0.
- Add 2 to both sides: x ≥ 2.
Answer: Domain: x ≥ 2
Vertex and range of a translated parabola
Describe the graph of y = (x − 3)^2 + 2 and state its range.
- Compare with y = (x − h)^2 + k: h = 3 and k = 2.
- The graph is y = x^2 moved 3 units right and 2 units up, with vertex (3, 2).
- The parabola opens upwards, so the lowest y-value is 2.
Answer: Vertex (3, 2); range y ≥ 2
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
If f(x) = 3x − 4, what is f(2)?
2 — f(2) = 3 × 2 − 4 = 6 − 4 = 2.
Which of these relations is not a function?
x^2 + y^2 = 9 — x^2 + y^2 = 9 is a circle. For example, x = 0 gives y = 3 and y = −3, so one input has two outputs.
What is the domain of f(x) = sqrt(x + 5)?
x ≥ −5 — We need x + 5 ≥ 0, which gives x ≥ −5.
What is the range of y = x^2 − 4?
y ≥ −4 — x^2 is never negative, so the smallest value of x^2 − 4 is −4 (when x = 0). The range is y ≥ −4.
How is the graph of y = (x + 2)^2 related to the graph of y = x^2?
Moved 2 units left — y = (x + 2)^2 = (x − (−2))^2, so the vertex moves to (−2, 0): 2 units to the left.
If f(x) = x^2 − 5x, find f(−2).
14 — f(−2) = (−2)^2 − 5 × (−2) = 4 + 10 = 14.
The function g(x) = 1/(x − 3) is defined for all real numbers except one value of x. What is that value?
x = 3 — When x = 3 the denominator is 0, and division by zero is undefined.
What is the minimum value of y for the function y = (x − 1)^2 + 6?
6 — (x − 1)^2 is never negative and equals 0 when x = 1, so the minimum value of y is 0 + 6 = 6.
Watch out for
- Thinking that a function cannot have two inputs with the same output. That is allowed; what is not allowed is one input with two outputs.
- Reading y = (x + 2)^2 as a shift to the right. Rewrite it as (x − (−2))^2, or test where the bracket equals zero.
- Forgetting brackets when substituting negatives, such as calculating −2^2 as −4 instead of (−2)^2 = 4.
- Confusing domain (inputs, x-values) with range (outputs, y-values).
Tips for parents
- Ask your learner to sketch every function before stating its range. A quick picture prevents most range errors.
- Graphing software or a graphics calculator can be a powerful way to check work and explore transformations, if one is available.
- Encourage your learner to explain in words what a function does, such as "square the input, then subtract 4".
Go further
- Graph y = x^2, y = (x − 3)^2, y = x^2 + 3 and y = −x^2 on the same axes. Write a rule describing each change.
- Find a real situation that is a function (such as postage cost by weight) and one that is not, and explain the difference.
- Investigate the domain and range of y = 1/x and describe what happens to the graph near x = 0.