A change outside the function acts vertically and behaves as written: f(x) + 3 moves up 3.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Inside acts horizontally, and backwards
f(x + 3) moves left 3, not right. The input must be 3 smaller to give the same output.
Stretches
2f(x) stretches vertically by 2. f(2x) compresses horizontally by 2 — again the reverse of what it looks like.
Order matters
Stretching then shifting is not the same as shifting then stretching.
Inside the bracket goes the wrong way
Changes inside the function act on the input and behave in reverse: f(x − 3) shifts right, and f(2x) compresses. Changes outside act on the output and behave as expected.
Why inside changes reverse
For f(x − 3) to produce the value f gave at x, the input must now be 3 larger, so the graph moves right. The reversal is a consequence of substitution, and thinking it through once removes the need to memorise it.
Order matters when combining
A horizontal stretch followed by a shift differs from the reverse. Factoring the inside into the form a(x − h) makes the correct order readable, which is exactly why that form is used.
Track the domain and range
Horizontal transformations change the domain; vertical ones change the range. Reporting the transformed domain and range is a routine exam requirement and follows mechanically from which kind was applied.
Step 2: Try It Yourself
Move the parabola around and watch which number does what.
Step 3: Watch an Example
One step at a time.
Watch Sam Apply Three Changes
Sam describes y = −2f(x − 1) + 5 in words.
- Step 1
The (x − 1) is inside, so it shifts right by 1.
Step 4: Your Turn
Practice makes it stick.
Inside the Bracket
Problem 1 of 2
f(x + 4) moves the graph which way? Answer 1 for left, 2 for right.
Outside the Bracket
Problem 2 of 2
f(x) − 6 moves the graph down by how much?
Which Way and How Far
1 of 8
f(x − 7). Right by how much?
2 of 8
f(x) + 2. Up by how much?
3 of 8
3f(x). Vertical stretch factor?
4 of 8
Match each transformation to its effect.
Tap a card on the left to start.
5 of 8
f(2x). Horizontal compression factor?
6 of 8
Order the steps for graphing y = 2f(x − 3) + 1 from the base graph.
- 1Stretch vertically by 2
- 2Shift up 1
- 3Shift right 3
7 of 8
The point (2, 5) is on y = f(x). What is its y on y = f(x) + 4?
8 of 8
The point (2, 5) is on y = f(x). What is its x on y = f(x − 3)?
Step 5: Quick Check
Show what you know.
Question 1 of 1
f(x + 9) moves the graph left by how much?
What You Learned
- Outside the function acts vertically and behaves as written.
- Inside acts horizontally and runs backwards.
- Several transformations together are order-dependent.