Part 25 of 30 · TransformationsSee the whole course
Stretching and squashing a graph: a·f(x) and f(bx)
A number in front, a·f(x), multiplies every height: bigger than 1 stretches the graph taller, between 0 and 1 squashes it flatter. A number on x, f(bx), acts sideways, and backwards: b bigger than 1 squeezes the graph toward the y-axis.
A number in front: stretch and squash
Multiplying the outside multiplies every output. Take f(x) = x² and put a 2 in front:
x: 0, 1, 2, 3
x²: 0, 1, 4, 9
2x²: 0, 2, 8, 18
The graph climbs twice as fast, so it is taller. That makes it look narrower, which surprises many people. A number between 0 and 1, like ½x², does the opposite: it flattens the bowl, so it looks wider. The vertex does not move, because 0 × anything is 0.
A number on x: sideways, and backwards
A number inside acts on x, so it acts sideways, and backwards. In y = (2x)² the graph reaches each height at half the old x, so it is squeezed toward the y-axis. A number below 1, as in (½x)², pulls the graph out wider.
(2x)² = 4x², so squeezing x² by 2 is the same as stretching it ×4.
√(4x) = 2√x, so squeezing √x by 4 is the same as stretching it ×2.
That is a special feature of these shapes. Do not rely on it for every function. Play with both numbers in the figure below.
- in front: 2, every height ×2, taller
- on x: 1, no change sideways
- point (1, 1) → (1, 2) · same as y = 2x²
All four moves together
For y = −2(x − 1)² + 3, read the moves from the inside out: slide 1 right, stretch by 2 and flip, lift 3 up. Check with a point: the parent point (1, 1) goes to x = 2, then −2 × 1 = −2, then +3 gives 1, so it lands on (2, 1). And −2(2 − 1)² + 3 = 1 ✓.
Sketch it fast
For y = a(x − h)² + k, start at the vertex (h, k). Then step 1 sideways and a up, then 2 sideways and 4a up. For y = 2(x − 1)² + 3: vertex (1, 3), then (2, 5) and (0, 5), then (3, 11) and (−1, 11).
Watch it explained in under a minute
The four mistakes to avoid
- Thinking a number bigger than 1 makes the graph wider. In front, it makes it taller, so it looks narrower.
- Mixing up in front and on x. 2x² multiplies heights. (2x)² acts on x.
- Forgetting that the number on x goes backwards. A 2 squeezes, it does not stretch.
- Doing the moves in the wrong order. Inside first, then the number in front, then the number at the end.
Try it yourself
Tap a question to see the answer.
1. Describe y = 3x² and say where it passes through.
It is x² stretched by 3: taller and narrower. The vertex stays at (0, 0), and it passes through (1, 3).
2. How does y = ½|x| compare with y = |x|? Does it pass through (4, 2)?
It is squashed to half the height, so it is flatter and wider. At x = 4: ½ × 4 = 2, so yes.
3. Find the moves, vertex and direction of y = −2(x + 1)² + 4.
Left 1, stretch 2 and flip, up 4. The vertex is (−1, 4) and it opens downward. It passes through (0, 2): −2(1)² + 4 = 2 ✓.
Where this fits in
That completes the four moves: shift, flip and stretch. y = a(x − h)² + k is exactly these moves applied to y = x², which makes it the vertex form of a quadratic. The next part of the course starts there, with what a quadratic function is.