Grade 10 · Grade 10: Polynomials, Models and Function Behaviour · Polynomials, Models and Function Behaviour
Transformations, Inverses and Composition
Shift, stretch and reflect graphs deliberately, then undo and chain functions.
Learning objectives
- Describe the effect of f(x) + k, k·f(x), f(kx) and f(x + k).
- Transform linear, quadratic and exponential graphs.
- Find inverses of simple functions.
- Interpret composite functions in context.
AERO Mathematics alignment
AERO.M10.AE.6
Identify the effect on the graph of a function of replacing f(x) by f(x) + k, k·f(x), f(kx) and f(x + k), and describe transformations of linear, quadratic and exponential graphs.
AERO.M10.AE.7
Find inverse functions for simple functions, understand composition of functions and interpret both in context.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
ENGAGE
Start here
You graph y = x², then y = (x + 3)² and then y = 2x².
Think about it
Predict how each graph moves or changes before you plot it.
Hint: Changes inside the bracket affect x, and they behave in the opposite way to what you expect.
EXPLAIN
Transforming, inverting and composing
| Change | Effect on graph | Example |
|---|---|---|
| f(x) + k | Vertical shift up by k | y = x² + 3 moves up 3 |
| f(x + k) | Horizontal shift LEFT by k | y = (x + 3)² moves left 3 |
| k·f(x) | Vertical stretch by k | y = 2x² becomes narrower |
| f(kx) | Horizontal compression by k | y = (2x)² compresses |
| −f(x) | Reflection in the x-axis | y = −x² opens downward |
Inside-the-bracket changes act on the input, so they behave opposite to intuition: adding 3 to x shifts the graph left, because the graph reaches the same output 3 units earlier.
Inverse functions
An inverse undoes the function. Swap x and y and solve: for f(x) = 3x − 7, write x = 3y − 7, so f⁻¹(x) = (x + 7)/3. The graph of an inverse is the reflection of the original in the line y = x.
Composition
(f ∘ g)(x) = f(g(x)) applies g first, then f. Unit conversion chains are compositions: convert miles to kilometres, then kilometres to metres.
Common misconception
f⁻¹(x) is not 1/f(x). The −1 marks the inverse function, not a negative exponent.
INVESTIGATION
Try it yourself
Pick one parent function and produce four transformed versions. Sketch each, write its equation and describe the transformation in words. Then find the inverse of one of them.
Watch
What Are Functions?
Math Antics · 10:00
Reinforces function notation before you transform graphs.
Before you watch: What does f(x) + 3 do to a graph?
- Describe the transformation from f(x) to 2f(x − 1).
- Find the inverse of f(x) = 3x − 7.
Interactive simulation · PhET
Graphing Slope-Intercept
Translate and stretch a graph, recording the equation change each time.
While you explore
- Which change moves the graph up, and which moves it right?
- Why does f(x + 3) shift left rather than right?
Key vocabulary
- Composition
- Doing one function and then another.
- Inverse function
- A function that undoes another one.
Practice questions
0/1 correct
Level 1 · Criterion A
How does the graph of y = (x + 4)² differ from y = x²?
MYP criterion tasks
Level 2 · Criterion C
Find the inverse of f(x) = 5x + 2 and verify it with one value.
Reflect & track
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