60-minute lesson
Year 11 Mathematics: Functions and their graphs
Lesson objective
- 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 criteria:
- 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.
Materials
- Grid paper, pencil and ruler
- A scientific or graphics calculator, or free graphing software if available
- Paper and pencil
Introduction
6 minIntroduce today's words:
- Function: A rule that gives exactly one output for each input.
- Function notation: Writing f(x) for the output of function f when the input is x. For example, if f(x) = 2x, then f(3) = 6.
- Domain: The set of all input values (x-values) for which a function is defined.
- Range: The set of all output values (y-values) that a function produces.
- Vertex: The turning point of a parabola, where it reaches its minimum or maximum value.
- Translation: A shift of a graph left, right, up or down without changing its shape.
Ask your child what they already know about functions and their graphs.
Explanation
12 minA 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
9 minEvaluating a function
If f(x) = 2x^2 − 3, find f(4) and f(−1).
Step 1: f(4) = 2 × 4^2 − 3 = 2 × 16 − 3 = 32 − 3 = 29.
Step 2: 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).
Step 1: The expression under a square root cannot be negative.
Step 2: So x − 2 ≥ 0.
Step 3: 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.
Step 1: Compare with y = (x − h)^2 + k: h = 3 and k = 2.
Step 2: The graph is y = x^2 moved 3 units right and 2 units up, with vertex (3, 2).
Step 3: The parabola opens upwards, so the lowest y-value is 2.
Answer: Vertex (3, 2); range y ≥ 2
Guided practice (do together)
12 min1. If f(x) = 3x − 4, what is f(2)?
(a) 2 (b) −2 (c) 10 (d) 6
2. Which of these relations is not a function?
(a) y = 2x + 1 (b) y = x^2 (c) x^2 + y^2 = 9 (d) y = 5
3. What is the domain of f(x) = sqrt(x + 5)?
(a) x ≥ 5 (b) x ≥ −5 (c) x > 0 (d) All real numbers
4. What is the range of y = x^2 − 4?
(a) y ≥ 0 (b) y ≥ 4 (c) All real numbers (d) y ≥ −4
Independent practice
15 min5. How is the graph of y = (x + 2)^2 related to the graph of y = x^2?
(a) Moved 2 units right (b) Moved 2 units up (c) Moved 2 units left (d) Moved 2 units down
6. If f(x) = x^2 − 5x, find f(−2).
7. The function g(x) = 1/(x − 3) is defined for all real numbers except one value of x. What is that value?
8. What is the minimum value of y for the function y = (x − 1)^2 + 6?
Questions to check understanding
- Can you use the vertical line test to decide whether a graph is a function?
- Can you evaluate f(a) for a given function and value a?
- Can you state the domain and range of linear, quadratic, square root and simple rational functions?
- Can you describe how y = (x − h)^2 + k is related to y = x^2?
- What was the trickiest part today?
Answer guide
1. 2 — f(2) = 3 × 2 − 4 = 6 − 4 = 2.
2. 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.
3. x ≥ −5 — We need x + 5 ≥ 0, which gives x ≥ −5.
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.
5. Moved 2 units left — y = (x + 2)^2 = (x − (−2))^2, so the vertex moves to (−2, 0): 2 units to the left.
6. 14 — f(−2) = (−2)^2 − 5 × (−2) = 4 + 10 = 14.
7. x = 3 — When x = 3 the denominator is 0, and division by zero is undefined.
8. 6 — (x − 1)^2 is never negative and equals 0 when x = 1, so the minimum value of y is 0 + 6 = 6.
Review
6 minRecap the success criteria together. Watch for these common misconceptions:
- 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).
Extension activities
- 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.
Suggested follow-up
- Revisit functions and their graphs tomorrow with two or three quick questions from memory.
- Try the online practice check for this topic and look at any questions that need another go.
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".