Function Transformer

Skill goal: understand translations, reflections, and scaling — see how a, h, and k in y = a·f(x−h) + k reshape any parent function.

Observe

The gray dashed curve is the parent function. Move the sliders and watch the blue curve — the same shape, shifted, flipped, or stretched.

Manipulate

Parent function:

Gray dashed: parent f(x) · Blue: y = a·f(x−h) + k · Teal dot: where (0, 0) moved

Predict

Before you change it, what do you think will happen? If h changes from 0 to 3, will the graph move left or right? Make a guess, then test it with the slider.

See

Explain

Why it works: h works inside the function, so it moves the graph the opposite way from what you might expect — h = 3 shifts the graph right, because x must be 3 larger to produce the same output. k is added at the very end, so it simply lifts the whole graph up (or down). a multiplies every output: |a| > 1 stretches the graph tall, |a| < 1 squashes it flat, and a negative a flips it across the x-axis.

Challenge

Keep learning

• Learn — lessons and videos
• Practice — interactive exercises
• Worksheets — printable PDFs
• Tools — more math labs