A change outside the function moves it vertically. f(x) + 3 lifts the whole graph by 3.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Inside the brackets
f(x − 3) shifts the graph 3 to the *right*, which surprises almost everyone the first time.
Why it feels backwards
To get the same output, x must now be 3 bigger. So the graph slides right, not left.
Stretching and reflecting
a·f(x) stretches vertically by a. A negative a also flips it upside down.
One set of rules for every curve
Replacing f(x) with f(x − h) shifts right by h; adding k outside shifts up by k; a multiplier outside stretches vertically; a negative reflects. These rules do not depend on what f is, which is why they are worth learning once.
Inside affects x, outside affects y
Changes inside the brackets act on the input and behave counter-intuitively — f(x − 3) moves right, not left. Changes outside act on the output and behave as expected. Sorting a transformation into inside or outside is the first step.
Why inside changes run backwards
For f(x − 3) to produce the value f used to give at x, the input must now be 3 larger — so the graph moves right. The reversal is a consequence of substitution, not an arbitrary rule, and thinking it through once fixes it.
Order matters when combining
A stretch followed by a shift is not the same as a shift followed by a stretch. When several transformations apply, work through them in the order the algebra dictates: inside changes right to left, outside changes as written.
Step 2: Try It Yourself
Change one parameter at a time and name what moved.
Step 3: Watch an Example
One step at a time.
Watch Ravi Describe y = −2(x − 3)² + 1
Ravi reads off every transformation from the equation.
- Step 1
The (x − 3) shifts the parabola 3 to the right.
Step 4: Your Turn
Practice makes it stick.
The Shift
Problem 1 of 2
y = (x − 4)². What is the x-coordinate of the vertex?
The Lift
Problem 2 of 2
y = x² + 7. What is the y-coordinate of the vertex?
Name the Move
1 of 4
y = (x + 5)². Vertex x-coordinate?
2 of 4
y = (x − 2)² + 6. Vertex y-coordinate?
3 of 4
y = −x². Does it open up or down?
4 of 4
y = 3x². Is it narrower or wider than y = x²?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y = (x − 6)² − 2. Vertex x-coordinate?
Question 2 of 2
Why does f(x − 3) shift the graph right?
What You Learned
- Changes outside the function move it vertically.
- Changes inside move it horizontally, and in the opposite direction.
- A multiplier stretches, and a negative one reflects.