A change outside the function moves the graph vertically; a change inside moves it horizontally, and backwards. A multiplier stretches; a negative one reflects. One set of rules for every curve.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Work out y = x² and y = (x − 2)² for x = 0, 1, 2, 3 and 4. Where does each graph reach its lowest point?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Outside is vertical
f(x) + 3 lifts every output by 3, so the whole graph rises 3. f(x) − 3 lowers it. The shape is untouched.
Inside is horizontal, and backwards
f(x − 3) shifts the graph 3 to the right. To get the output f used to give at x, the new input must be 3 bigger. So every point slides right, not left.
Stretching
a·f(x) multiplies every output by a: a = 2 doubles heights, a = ½ halves them. f(2x) squeezes horizontally by 2, because the input reaches every value twice as soon.
Reflecting
−f(x) flips the graph across the x-axis: every output changes sign. f(−x) flips it across the y-axis.
The order of moves
Read from the inside out: shift inside first, then stretch, then reflect, then shift outside. y = −2(x − 3)² + 1: right 3, stretch 2, flip, up 1. Vertex at (3, 1).
Track one point
Follow the vertex of the parent curve through the moves. For y = |x|, the corner (0, 0) under y = 3|x + 2| − 4 goes to (−2, −4), and the arms are three times as steep.
Step 2: Try It Yourself
Tap and try it out.
Describe y = 3|x + 2| − 4, starting from y = |x|
- 1y = |x|: corner at (0, 0)the parent
Step 0 of 4
Step 3: In Real Life
Noise-cancelling headphones
A sound wave stretched taller is louder; shifted along, delayed; flipped, inverted to cancel noise. Noise-cancelling headphones apply exactly these transformations to a curve, thousands of times a second.
Step 4: Watch an Example
One step at a time.
Watch Maya Describe y = −3(x + 1)² + 5
A fresh equation, read from the inside out, with the vertex of y = x² tracked through every move and a check at the end.
- Step 1
Reading from the inside out, the bracket comes first. (x + 1) is inside the function, so it moves the graph horizontally, and backwards: plus 1 means left 1. To give the output x² used to give at 0, the new input has to be −1. The vertex slides from (0, 0) to (−1, 0).
Watch Leo Write the Equation
The moves are given in words: y = x², shifted left 2, stretched vertically by 3, reflected across the x-axis, then up 4. Leo writes the first two moves, and you write the last two.
- Step 1
Leo builds the equation in the reading order, inside out. Left 2 is a horizontal move, so it goes inside the bracket, and backwards: left 2 is written (x + 2). So far, y = (x + 2)², with the vertex at (−2, 0).
Watch Ana Track a Point
The point (2, 5) is on the graph of y = f(x). Ana finds where it lands on y = 2f(x − 3) + 1 without knowing what f is.
- Step 1
Nothing about f is known except one fact: f(2) = 5. Every move changes that point in a way that does not depend on f, so the point can be tracked on its own.
Step 5: Your Turn
Practice makes it stick.
The Stall Fee
Problem 1 of 2
A market stall's daily profit, in dollars, is p(x) = −2(x − 6)² + 200, where x is the price of one item in dollars. The market raises its daily stall fee by $40, so the new profit is p(x) − 40. What is the new highest daily profit, in dollars?
dollars
Twice the Traffic
Problem 2 of 2
A café's weekly profit, in dollars, is p(x) = −5(x − 4)² + 300, where x is the price of a coffee in dollars. A second café on a busier street is twice the size in every way — twice the sales and twice the costs at every price — so its profit is q(x) = 2p(x). What is the highest weekly profit of the second café, in dollars?
dollars
Read the Equation
Inside out: the bracket first, then the multiplier, then the sign, then what is added outside.
1 of 5
y = (x + 3)² − 5. What is the x-coordinate of the vertex?
2 of 5
y = −(x − 2)² + 7. Where is the vertex, and which way does the curve open?
3 of 5
Write the equation of y = x² shifted right 5 and down 2. Type the right-hand side.
4 of 5
y = |x| stretched vertically by 2, then shifted left 6. Which equation?
5 of 5
The point (3, 9) is on y = x². Where does it land on y = 2(x − 1)² + 4?
Whose Step?
One line in each reading is a slip. Find it, then fix it.
1 of 4
Priya read y = (x + 4)² − 1, found the vertex (4, −1), then expanded it. One line is a slip. Which?
Priya's reading of y = (x + 4)² − 1
- 1y = x²: vertex at (0, 0)
- 2(x + 4)²: vertex at (4, 0)
- 3(x + 4)² − 1: vertex at (4, −1)
- Answery = x² + 8x + 15
2 of 4
What is the x-coordinate of the vertex, correctly?
3 of 4
Dev wrote an equation for y = x² stretched vertically by 2 and shifted up 3. One line is a slip. Which?
Dev's equation for y = x², stretched by 2 and up 3
- 1y = x²: vertex at (0, 0)
- 2y = 2x²: vertex at (0, 0)
- 3y = 2(x² + 3): vertex at (0, 6)
- Answery = 2x² + 6
4 of 4
Write the equation Dev should have reached. Type the right-hand side.
Pick the Move
Decide which move, or which order, does the job. No tracking all the way through is needed.
1 of 2
Ana and Leo both describe y = 2(x − 1)² + 3 from y = x². Ana: right 1, stretch by 2, up 3. Leo: up 3, stretch by 2, right 1. Whose order lands the vertex in the right place?
2 of 2
You want y = |x| flipped so it opens downward, with its corner still at (0, 0). Which move does it?
Same Graph?
Judge without plotting: which of these draw what the question says?
1 of 2
Select every equation whose vertex is at (−3, 2).
2 of 2
Select every pair that draws the same graph.
Step 6: 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?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
For which number a does y = a(x − 2)² + 3 pass through the point (4, 11)?
Stretch 2 of 2
Using each of the numbers 2, 3 and 4 exactly once for a, h and k in y = a(x − h)² + k, write an equation whose vertex lies on the line y = 2x. Type the right-hand side.
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.
- Read inside out, and track one point through the moves.
For the grown-up
The misconception this lesson targets is reading the bracket forwards: (x + 4) looks like "add 4", so the graph "must" move right 4, when it moves left. Its cousin is treating the four moves as if their order did not matter, so an up-3 written before a stretch by 2 gets doubled. The equation itself gives the order: whatever is done to x inside happens first, the multiplier and the sign next, and the number added outside last. A check with the vertex’s x-coordinate settles any doubt in one line.