Transforming graphs
What this lesson covers
- Adding outside moves a graph up or down.
- Changing x inside the bracket moves it the opposite way.
- A multiplier stretches, and a negative reflects.
The four moves
- Start from y = f(x).
| New graph | What happens |
|---|---|
| y = f(x) + a | Up a |
| y = f(x − a) | Right a |
| y = −f(x) | Reflected in the x-axis |
| y = k f(x) | Stretched vertically by k |
From y = x² to y = 2(x − 3)² + 1
- Read the moves from the numbers.
- right 3x − 3
- stretched by 2the multiplier
- up 1the + 1
- vertex (3, 1)
- y-intercept = 2(9) + 1 = 19
Common mistakes
- Moving y = (x − 3)² left by 3.x − 3 moves it right by 3, because x has to be 3 to give zero.
- Forgetting the stretch when finding the y-intercept.Substitute x = 0 into the whole equation.
Your turn
- 1. Describe how y = −(x + 1)² + 4 is made from y = x².
- 2. Find its vertex.
- 3. Find its x-intercepts.
Show answer
1. Left 1, reflected in the x-axis, up 4 2. (−1, 4) 3. x + 1 = ±2, so x = 1 and x = −3
What to take away
- Outside the bracket acts as written.
- Inside the bracket acts in reverse.
More in AS 91257
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