Parent Functions & Transformations

A handful of basic shapes, moved and stretched, account for almost every graph you will meet.

Trace a parabola onto tracing paper, then slide the paper.

Nothing about the curve changed. It is the same width, the same shape, the same everything.

one curve, traced and moved slid up and left squashed same shape — only its place and width changed

All that changed is where it sits. Two numbers describe that: how far across, how far up.

Now pull the paper wider without moving it. Still the same curve, just stretched — one more number.

So a graph needs two things: which family it belongs to, and what was done to it. y = a(x-h)2 + k is exactly that sentence written in symbols.

Changing b slides a line; m keeps it parallel change b and the line slides; change m and it tilts y = x + 2 y = x y = x - 3 same m, so all three stay parallel f(x) + k moves the graph up k; f(x + k) moves it left k, which surprises nearly everyone

The families worth knowing by sight

Six shapes cover most of the course: the line, the parabola x2, the cubic x3, the absolute value |x|, the root √x and the reciprocal 1x.

Learn what each one looks like at rest, and every later graph becomes one of them, moved.

Sketch each from three points if you are unsure — the shape returns faster than the formula does.

Why these six

They are the graphs you cannot build from the others. Everything else in Algebra II is one of these six with numbers attached.

Outside the function moves it vertically

Anything done outside f happens to the output, so it affects the graph vertically and does what it says.

f(x) + 3 lifts the graph three; -f(x) flips it upside down; 2f(x) stretches it to twice the height.

These are the intuitive ones, because the arithmetic happens after the function has already done its job.

Outside is honest

Add three outside, the graph rises three. What you write is what you get — vertically.

Inside the function moves it horizontally, backwards

Anything done inside the bracket happens to the input, and the graph responds in the opposite direction.

f(x - 3) moves the graph three to the right, not left. To get the output f used to give at 0, you must now feed in 3.

Likewise f(2x) squashes the graph to half its width, because the input reaches its usual values twice as fast.

The one everybody gets backwards

Ask what x makes the bracket zero. In f(x-3) that is x = 3, so the graph’s starting point has moved to 3. The bracket tells you the answer directly.

Reading a transformed formula

In y = a f(x - h) + k, read h and k as the new home of the parent point, and a as the stretch.

For y = -2(x+1)2 + 5: the vertex sits at (-1, 5), the graph is twice as tall and opens downward.

Order matters when you apply them by hand: stretch and flip first, then translate.

Check with one point

Put the parent’s key point through the transformation and see where it lands. If the vertex of (x+1)2 + 5 does not come out at (-1,5), a sign slipped.

Piecewise functions

Some situations need different rules on different stretches: a parking charge that is flat for two hours, then hourly.

Write each rule with the interval it governs, and evaluate by first asking which piece the input falls in.

On the graph, a filled dot means the endpoint is included; an open dot means it is not.

One input, one piece

The intervals must not overlap, or a single input would have two outputs — and then it is not a function at all.

Worked examples

Describing a transformation

Describe how y = -12(x - 4)2 + 3 relates to y = x2.

  1. The bracket is (x-4), so the graph moves 4 right. Inside the function, and the opposite of the sign shown.
  2. The -12 flips it downward and squashes it to half height. Outside the function, so vertical and honest.
  3. The +3 lifts it 3. The vertex lands at (4, 3). Outside again, so straight up.

Right 4, reflected down, half as tall, up 3; vertex (4,3)

Writing the formula from a picture

An absolute value graph has its corner at (-2, 1) and opens upward, twice as steep as y = |x|. Write its equation.

  1. Corner at x = -2 means the bracket is (x + 2). The bracket must be zero at the corner.
  2. Corner at y = 1 means +1 outside. Vertical shift is exactly the y-value.
  3. Twice as steep is a factor of 2 in front. Positive, since it still opens upward.

y = 2|x + 2| + 1

Evaluating a piecewise function

For f(x) = 2x + 1 when x < 3, and f(x) = x2 when x ≥ 3, find f(2) and f(3).

  1. 2 < 3, so f(2) uses the first rule: 2(2) + 1 = 5. Choose the piece before doing any arithmetic.
  2. 3 ≥ 3, so f(3) uses the second rule: 32 = 9. The ≥ puts the endpoint 3 in the second piece.
  3. The two rules disagree at 3 — the graph jumps. That is allowed; it is still one output per input.

f(2) = 5, f(3) = 9

Words to know

Parent function
The plainest member of a family — y = x2, y = |x|, y = √x, y = 1x — before anything is done to it.
Translation
A slide. f(x) + k moves the graph up; f(x - h) moves it right.
Reflection
A flip. -f(x) flips across the x-axis; f(-x) flips across the y-axis.
Vertical stretch
Multiplying the whole output: a f(x) pulls the graph away from the x-axis when |a| > 1.
Piecewise function
A function defined by different rules on different stretches of the domain.

Mistakes to watch for

Shifting horizontally the wrong way. f(x - 3) moves the graph right, not left. Ask which x makes the bracket zero: for (x-3) that is 3, so the graph now starts there.

Treating an inside stretch as a vertical one. f(2x) halves the width; 2f(x) doubles the height. They look alike in print and do opposite things, so check whether the 2 is inside the bracket.

Applying the shift before the stretch. In y = 2(x-1)2 + 3 the stretch acts on the parent, then the shift moves the result. Shifting first moves the wrong curve and lands the vertex in the wrong place.

Squaring the coefficient by accident. In y = (2x)2 the 2 is inside and gets squared to 4; in y = 2x2 it is outside and stays 2. The graphs differ by a factor of two in height.

Using the wrong piece at a boundary. If the definition says x ≥ 3, then 3 belongs to that piece. The inequality sign at the endpoint is the whole instruction, and a filled or open dot on the graph says the same thing.

Graphic designers call this the same thing mathematicians do: a transform. Moving, scaling and flipping a shape are the only three operations most design tools offer.

The reason y = f(x - 3) shifts the graph right and not left catches almost everyone. The graph does whatever it must so the function still sees its usual input.

Every animation you have watched is a transformation applied over and over, a fraction of a second apart.

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