Math · Functions

Graph Transformations

Simulation loading

Open Model Lab is preparing the live lab, controls, and graph surface for this concept.

Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Rational Functions / Asymptotes and BehaviorSame graph-reading habit

Key takeaway

Inside changes move where you read x before the graph computes y.
Outside changes shift or rescale the y-values after the input is chosen.
A reflection changes the side of the axis where the same landmark appears, not just the position of the whole graph.

Common misconception

A positive change inside the brackets is not just a left-right slogan; it is a change in which x-value feeds the function.

Do not guess from the sign alone. Track a known base landmark and ask where the inside input must equal that same base x-value again.

Keep this idea moving

Open the next concept, route, or track only when you want the current model to widen into a larger branch.

Carry the same graph-shift language into vertical and horizontal asymptote behaviorFunctions

Rational Functions / Asymptotes and Behavior

Vary one shifted reciprocal family so domain breaks, vertical and horizontal asymptotes, intercepts, and removable-hole behavior stay tied to the same graph.

Keep the graph-first branch moving into growth, decay, and target timeFunctions

Exponential Change / Growth, Decay, and Logarithms

Change one starting value, one rate, and one target so growth, decay, doubling or half-life, and logarithmic target time all stay tied to the same live curve.

Useful comparisonCalculus

Derivative as Slope / Local Rate of Change

Slide a point along one curve, tighten a secant into a tangent, and connect local steepness to the derivative graph without leaving the same live bench.

Stay with this concept

Review, test, or free-play here when you want one more same-concept pass.

More practice optionsReview on the bench · Worked examples

Practice optionReview on the benchReplay the guided setup with the key controls and graphs in sight.
Practice optionWorked examplesCompare against solved walkthroughs after you try the setup yourself.
Practice optionFree play on the benchExperiment freely with the live controls before moving to another concept.

Reference and support

Use these calmer sections when you want a fuller explanation, examples, or accessibility support after the guided flow.

Return here when you want the slower reference pass.

Open reference and support

Equation snapshotTransformation rule · Transformed x landmarkShow 2 equations

Use one landmark rule for x and one for y so the live graph stays grounded in visible motion.

Transformation rule
$g (x) = a f (m (x - h)) + k$
Shows one base graph being changed by an inside input move, an outside vertical scale, and an outside vertical shift.
Transformed x landmark
$x_{v} = h + m x_{0}$
Tracks where a known base x-landmark lands after the inside move and the horizontal placement.

Why it behaves this way

Explanation

Graph transformations make more sense when one reference graph stays visible and every parameter acts on that same shape. This module keeps the base curve on screen while horizontal shift, vertical shift, vertical scale, and optional y-axis reflection build the transformed graph in the same coordinate plane.

The safest reading is landmark-first. Pick a known point on the base graph, such as the vertex, then track how its x-location changes under the inside move and how its y-value changes under the outside move. That is what separates horizontal placement from vertical stretching or reflection.

Key ideas

01Inside changes decide where a chosen base x-landmark ends up on the transformed graph.

02Outside changes act on y-values after the horizontal placement has been determined.

03A negative vertical scale reflects the graph across the x-axis, while the optional inside reflection reverses left-right orientation across the y-axis before the horizontal shift places the graph.

Quick test

Loading saved test state.

Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

A coordinate grid shows the base graph and the transformed graph together. Sliders and a reflection toggle change the horizontal shift, vertical shift, vertical scale, and optional y-axis reflection, while vertex markers keep the main landmark visible.

A readout card reports the current values of $h$ , $k$ , $a$ , the transformed vertex, and the y-intercept so the graph and the algebra can be checked together.

Graph summary

The first graph tab compares the base and transformed curves on one set of axes. The second graph tab shows how the current vertical shift changes the transformed vertex height as the vertical scale varies.

These graph views stay tied to the same landmark story: one view shows where the graph is, and the other shows how the transformed vertex height responds to $a$ .

Bench tools and share links

Keep stable concept links and exact-state sharing tucked away until you actually need to relaunch or share the bench.

Try this setup

Jump to a named bench state or copy the one you are looking at now. Shared links reopen the same controls, graph, overlays, and compare context.

Featured setups

Stretch upward

Open the stretched graph with the base curve and transformed vertex visible together.

Reflect, then shift right

Open the reflected case and track the base vertex as the graph flips left-right and then moves.

Saved setups

Saved setups are a Supporter study tool. Stable concept links still work for everyone.

Checking saved setup access

Open Model Lab is resolving whether this bench can save locally, sync to an account, or open Supporter-only compare tools.

Copy current setup

Exact-state sharing is part of Supporter. Stable concept and section links still stay available.

Stable links

Progress and next steps

Keep progress signals, starter-track handoffs, and review prompts available without letting them compete with the live lesson flow.

Progress

Loading progress

Loading saved concept progress for this browser or synced account before showing completion status.

Starter track

Step 1 of 6

Functions and Change

Rational Functions / Asymptotes and Behavior is what this track opens next.

This concept is the track start.

See next concept