HomeConcept libraryFunctionsFunctions and Change

MathFunctionsIntroStarter track

Concept module

Graph Transformations

Move one parent curve with honest controls so shifts, vertical scale, and reflections stay tied to the same overlaid graph and landmark points.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Graph transformations

Drag the highlighted vertex to reposition the transformed graph.

Graphs

Switch graph views without breaking the live stage and time link.

Function graph

Compare the reference curve with the transformed curve on the same axes.

x: -4.64 to 4.64y: -4 to 25

Base curveTransformed curve

Hover or scrub to link the graph back to the stage.x / y

Controls

Adjust the physical parameters and watch the motion respond.

Horizontal shift

h

Move the transformed graph left or right.

Vertical shift

k

Move the transformed graph up or down.

Vertical scale

a

Stretch, compress, or reflect the graph across the x-axis.

Reflect across y-axis

m

Flip the inside input left-to-right before the horizontal shift acts.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 3

Change only the horizontal shift and watch the transformed vertex slide along one level curve before any height changes.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

h

Horizontal shift

Moves the transformed graph left or right by changing where the inside landmark condition is satisfied.

Graph: Function graphOverlay: Vertex markersOverlay: Shift guide

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use these prompts when the signs start to feel slippery. Each one points back to a visible landmark on the graph.

ObservationPrompt 1 of 3

Graph: Function graph

Change only the horizontal shift and watch the transformed vertex slide along one level curve before any height changes.

Control: Horizontal shiftGraph: Function graphOverlay: Vertex markersOverlay: Shift guideEquation

h

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

2 visible

Overlay focus

Reference curve

Keep the original graph visible.

What to notice

The transformed graph is always being compared against the same base shape.
The y-axis reflection is easiest to see when the reference graph stays on screen.

Why it matters

A stable reference graph makes the parameter effects readable instead of abstract.

Control: Horizontal shiftControl: Vertical scaleControl: Reflect across y-axisGraph: Function graphEquation

m

Equation

h

Challenge mode

Use the graph like a landmark-matching bench: keep the reference curve visible, then place one reflected and scaled vertex honestly instead of guessing from symbols.

0/1 solved

MatchCore

4 of 8 checks

Reflect and land the vertex

Build a reflected graph whose transformed vertex lands near

(- 1, 2.4)

. Keep the reference curve and shift guide on so the move stays tied to the base landmark.

Graph-linkedGuided start2 hints

Suggested start

Start from the base graph, then use the reference curve, vertex markers, and shift guide together.

Matched

Open the Function graph graph.

Function graph

Matched

Keep the Reference curve visible.

Matched

Keep the Vertex markers visible.

Pending

Keep the Shift guide visible.

Off

Matched

Keep the horizontal shift close to

0

so the leftward landing comes from the reflection, not a big slide.

Pending

Make the vertical scale negative with size about

1.2

Pending

Land the transformed vertex x-value between

- 1.05

and

- 0.95

Pending

Land the transformed vertex y-value between

2.25

and

2.55

-2

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

The transformed curve stays centered horizontally, stays centered vertically, and keeps the original vertical scale. The inside input keeps its left-right orientation. The transformed vertex sits near (1, -2).

Equation detailsDeeper interpretation, notes, and worked variable context.

Transformation rule

Keeps one reference graph visible while the inside change, vertical scale, and vertical shift act on the same base shape.

g (x) = a f (m (x - h)) + k

h

Horizontal shift 0

k

Vertical shift 0

a

Vertical scale 1

m

Inside reflection Off

Transformed x landmark

Tracks where a known x-location from the base graph lands after the inside change.

x_{v} = h + m x_{0}

h

Horizontal shift 0

m

Inside reflection Off

Transformed y landmark

Tracks how the vertical scale and vertical shift move a known output value.

y_{v} = a y_{0} + k

k

Vertical shift 0

a

Vertical scale 1

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 1 compact task ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Stable concept and section links stay public below while exact-state setup sharing stays behind premium.

Premium

Exact-state setup sharing

Stable concept and section links stay available below. Premium copies the live bench state, compare labels, overlays, inspect time, and public experiment-card payload into reusable setup links.

You can sign in if you want this tied to an account later, but signing in alone does not upgrade the account.

View plans Sign in

Stable links

Starter track

Step 1 of 30 / 3 complete

Functions and Change

Next after this: Derivative as Slope / Local Rate of Change.

1. Graph Transformations2. Derivative as Slope / Local Rate of Change3. Integral as Accumulation / Area

This concept is the track start.

See next concept

Short explanation

What the system is doing

Graph transformations become easier to trust when one reference graph stays visible and every parameter change acts on that same shape. This module keeps the base curve on screen, then lets horizontal shift, vertical shift, vertical scale, and optional reflection across the y-axis reshape it in one live coordinate plane.

The goal is not to memorize a slogan about moving left or right. The point is to watch where a landmark on the base graph lands after the inside change, the outside change, and the vertical scale have all acted together.

Key ideas

01The horizontal shift changes where the graph sits on the x-axis before the outer vertical scale and vertical shift are applied to the y-values.

02The vertical scale changes every output value at once. A negative value reflects the graph across the x-axis, and a magnitude below one compresses it.

03Reflecting the inside input across the y-axis reverses the left-right orientation of the base graph before the horizontal shift places it.

Worked example

Read the full frozen walkthrough.

Frozen walkthrough

Use the current transform rather than a detached worksheet. The same controls drive the stage, the graph tab, and these substitutions.

Live worked examples are available on Premium. You can still read the full frozen walkthrough on the free tier.

View plans

Frozen valuesUsing frozen parameters

The base graph has a vertex at $(1, - 2)$ . For the current transform, where does that landmark land?

h

Horizontal shift

k

Vertical shift

a

Vertical scale

1. Apply the inside change to the base x-location

The base vertex starts at

x_{0} = 1

. After the optional y-axis reflection, the inside landmark is at

1

2. Place that landmark with the horizontal shift

The transformed vertex x-coordinate is

x_{v} = h + 1 = 0 + 1 = 1

3. Apply the vertical scale and shift to the base y-value

The transformed vertex y-coordinate is

y_{v} = a (- 2) + k = 1 (- 2) + 0 = - 2

Current transformed vertex

(x_{v}, y_{v}) = (1, - 2)

Without the y-axis reflection, the horizontal shift simply translates the base vertex left or right before the vertical change is applied.

Common misconception

A positive horizontal shift should always move the graph to the left because the sign inside the brackets looks backwards.

The clean way to read it is to track where a known landmark on the base graph must end up after the inside input is satisfied.

In this module the base vertex starts at $x = 1$ . Solving the inside expression for that same landmark shows where the transformed vertex actually lands.

Mini challenge

Set the graph so the transformed vertex is left of the y-axis but above the x-axis, and make sure the graph has been reflected across the x-axis.

Prediction prompt

Decide which controls must change sign before you try it.

Check your reasoning

You need the transformed vertex x-coordinate to be negative, the transformed vertex y-coordinate to be positive, and the vertical scale to be negative so the graph reflects across the x-axis.

The horizontal location depends on the inside reflection plus the horizontal shift, while the height depends on the vertical scale and the outside vertical shift. Reflection across the x-axis comes from the sign of the vertical scale.

Quick test

Variable effect

Question 1 of 3

Reason from the live graph and the landmark formulas together. These checks are about where the graph goes, not just symbol matching.

Which change moves the graph up without changing its opening or left-right position?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

A coordinate grid shows the base graph and the transformed graph together. The transformed vertex can be dragged directly, while sliders and a reflection toggle change the horizontal shift, vertical shift, vertical scale, and y-axis reflection.

A readout card reports the current values of h, k, a, the transformed vertex, and the y-intercept.

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.

Prediction mode, compare mode, and guided overlays all stay tied to these same graph relationships.

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Local slope grows out of reading how a graph changesCalculus

Graph Transformations

See each variable before you move it.

Reference curve

Reflect and land the vertex

Functions and Change

What the system is doing

Read the full frozen walkthrough.

The base graph has a vertex at (1,−2). For the current transform, where does that landmark land?

Which change moves the graph up without changing its opening or left-right position?

Keep moving through the concept map.

Derivative as Slope / Local Rate of Change

Integral as Accumulation / Area

Vectors in 2D

The base graph has a vertex at $(1, - 2)$ . For the current transform, where does that landmark land?