Parent functions
A parent function is the simplest form of a family of graphs. Learn their shapes by heart:
- y = x (straight line through the origin)
- y = x² (U-shaped parabola, vertex at (0, 0))
- y = x³ (S-shaped cubic)
- y = |x| (V shape)
- y = √x (half a sideways parabola, starts at (0, 0))
- y = 1/x (two curves near the axes; the axes are asymptotes)
Every graph in this lesson is one of these, moved, stretched or flipped.
Translations: moving the graph
Vertical: y = f(x) + k moves the graph up k units (down if k is negative). Each output just gets k added.
Horizontal: y = f(x − h) moves the graph right h units (left if h is negative). This feels backwards. Think of it like this: the new graph must reach the same height at x = h that the old graph reached at x = 0. So it is pushed right.
Example: y = (x + 4)² − 1 is y = x² moved left 4 and down 1. Its vertex is at (−4, −1).
Some books write a translation as a vector: moving by (h, k) means right h and up k.
Stretches and reflections
Vertical stretch: y = a·f(x) multiplies every y-value by a. If a = 3, the graph is 3 times as tall. If 0 < a < 1, it is squashed towards the x-axis. Points on the x-axis do not move.
Horizontal stretch: y = f(bx) multiplies every x-value by 1/b. If b = 2, the graph is squeezed to half its width. If b = ½, it is twice as wide. Points on the y-axis do not move.
Reflections: y = −f(x) flips the graph in the x-axis (upside down). y = f(−x) flips it in the y-axis (left ↔ right).
Even functions (like x² and |x|) look the same after f(−x). Odd functions (like x³) give the same graph for −f(x) and f(−x).
Combining transformations
The general form is y = a·f(b(x − h)) + k. A safe order for drawing it:
- Horizontal stretch by 1/b (and flip in the y-axis if b < 0).
- Vertical stretch by a (and flip in the x-axis if a < 0).
- Translate right h and up k.
To find where a point goes, use: (x, y) → (x/b + h, a·y + k). For example, on y = 2(x − 1)² + 3, the parent point (1, 1) goes to (1 + 1, 2·1 + 3) = (2, 5).
Watch out: y = f(2x − 6) is not a shift of 6. Factor first: f(2(x − 3)) means squeeze by ½, then move right 3.
Try it: predict, then check
Before you touch the sliders, write down where the vertex of y = −(x + 1)² + 2 will be and whether it opens up or down. Then set a = −1, h = −1, k = 2 in the free-play step. Were you right? Now try y = |2x| and y = 2|x|. Are they the same graph? (They are! Can you explain why?)
Key formulas and definitions
- y = f(x) + k → up k
- y = f(x − h) → right h
- y = a·f(x) → vertical stretch by a; a < 0 flips in x-axis
- y = f(bx) → horizontal stretch by 1/b; b < 0 flips in y-axis
- y = a·f(b(x − h)) + k: point (x, y) → (x/b + h, a·y + k)
Worked examples
1. Describe how y = (x − 5)² + 2 is made from y = x².
x − 5 inside → move right 5. +2 outside → move up 2. Vertex goes from (0, 0) to (5, 2).
2. Describe y = −3|x| from y = |x|.
Multiply outside by −3: vertical stretch by 3, then flip in the x-axis. The V becomes a narrower upside-down V, vertex still at (0, 0).
3. Write the equation when y = √x is moved left 4 and down 1.
Left 4 → replace x with x + 4. Down 1 → subtract 1. y = √(x + 4) − 1. It now starts at (−4, −1).
4. The point (2, 8) is on y = f(x). Where is it on y = f(2x)?
Inside ×2 squeezes x by ½: (2, 8) → (1, 8). Check: at x = 1, f(2·1) = f(2) = 8. ✓
5. Find the vertex of y = 2(x + 3)² − 4 and say if it opens up or down.
Compare with a(x − h)² + k: a = 2, h = −3, k = −4. Vertex (−3, −4). a > 0, so it opens up, and it is narrower than y = x².
6. Describe y = f(3x − 6) + 1 as a list of transformations.
Factor inside: 3x − 6 = 3(x − 2). So y = f(3(x − 2)) + 1: horizontal stretch by 1/3, then right 2, then up 1. A point (x, y) goes to (x/3 + 2, y + 1).
Common mistakes
- Moving f(x − 3) to the left. A minus inside moves the graph RIGHT.
- Thinking f(2x) makes the graph twice as wide. It squeezes it to half the width.
- Reading f(2x − 6) as a shift of 6. Factor first: f(2(x − 3)) is a shift of 3.
- Mixing up −f(x) (flip upside down, in the x-axis) and f(−x) (flip left–right, in the y-axis).