Functions & Their Graphs

Graphing Transformations

Shifting, stretching, and reflecting a parent function's graph.

Intermediate20 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Why a Horizontal Shift Feels Backwards

y = x² has its vertex at (0, 0). Before reading on, predict the vertex of y = (x − 3)² + 2 — and notice that the −3 shifts the graph right, even though it looks like it should mean 'move left 3,' the way −3 usually would.

Definition — Transformation of a Parent Function

For y = a · f(b(x − h)) + k, applied to a parent function f: h shifts horizontally, k shifts vertically, |a| stretches or compresses vertically (and a < 0 reflects over the x-axis), and |b| stretches or compresses horizontally (and b < 0 reflects over the y-axis).

The horizontal shift feels backwards because the transformation happens inside the function, before f is ever evaluated. The parent function y = x² reaches its vertex when its input is exactly 0. For y = (x − 3)² to reach that same 'input is 0' condition, x itself has to equal 3, not −3 — the graph needs to shift right to compensate for the subtraction sitting inside the parentheses. The shift direction is forced by algebra, not an arbitrary rule to memorize.

Worked Example — Describing Transformations from an Equation

Describe the transformations in y = −2(x + 1)² − 4, compared to the parent y = x². Rewrite x + 1 as x − (−1) to match the standard form: h = −1 (shift left 1), k = −4 (shift down 4), a = −2 (vertical stretch by a factor of 2, and reflected over the x-axis since a is negative). The vertex moves from (0, 0) to (−1, −4), and the parabola opens downward.

Worked Example — Writing an Equation from a Description

Write an equation for the parent function y = √x, shifted right 4, stretched vertically by a factor of 3, and reflected over the x-axis. Apply each transformation to the parent form: y = −3√(x − 4).

Function Explorer

f(x) =

sin(x)

Transform: g(x) = a·f(b(x − h)) + k

a — vertical stretch/reflect1

b — horizontal stretch/reflect1

h — horizontal shift0

k — vertical shift0

Composition

Compose with another function

Trace mode

Show inverse view

Analysis (of the transformed function, in view)

y-intercept: (0, 0)
x-intercepts: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema: local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-1, 1]

Tip

Read a plus or minus sign inside the function's argument (affecting x directly) as the opposite of its usual direction, and a plus or minus sign outside the function (added at the very end) as its usual, expected direction — that's the one consistent rule underlying every horizontal-vs-vertical shift.

Common Mistakes

Shifting left instead of right (or vice versa) for a horizontal transformation, treating (x − h) the same intuitive way as an ordinary subtraction outside the function.
Rewrite the expression inside the parentheses as x minus a specific number first — a subtraction inside shifts right, and an addition inside (equivalent to subtracting a negative) shifts left.
Applying a horizontal stretch/compression using b directly as the scale factor, instead of 1/b.
For y = f(bx), the graph compresses horizontally when |b| > 1 and stretches when |b| < 1 — the actual horizontal scale factor is 1/|b|, the reciprocal of b, not b itself.

Key Takeaways

A parent function's graph can be shifted, stretched, compressed, and reflected using y = a · f(b(x − h)) + k.
Horizontal shifts move opposite to the sign inside the function, since the transformation happens to the input before evaluation.
Vertical changes (a and k) behave in the expected, intuitive direction, since they happen to the output after evaluation.

Summary

Transformations give precise control over a parent function's graph, once the backwards-feeling horizontal direction is understood algebraically. The next lesson looks at a special kind of built-in symmetry some functions have on their own, without any transformation applied.

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