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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.

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Featured setups

Stretch upward

Open the stretched graph with the transformed vertex and the reference curve visible.

Reflect and shift right

Open the reflected case and watch the vertex move after the inside change.

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Starter track

Step 1 of 60 / 6 complete

Functions and Change

Next after this: Rational Functions / Asymptotes and Behavior.

1. Graph Transformations2. Rational Functions / Asymptotes and Behavior3. Exponential Change / Growth, Decay, and Logarithms4. Derivative as Slope / Local Rate of Change+2 more steps

This concept is the track start.

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Why it behaves this way

Explanation

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.

Frozen walkthrough

Step through the frozen example

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

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View plans
Frozen valuesUsing frozen parameters

The base graph has a vertex at . For the current transform, where does that landmark land?

Horizontal shift

0

Vertical shift

0

Vertical scale

1

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

The base vertex starts at . After the optional y-axis reflection, the inside landmark is at .

2. Place that landmark with the horizontal shift

The transformed vertex x-coordinate is .

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

The transformed vertex y-coordinate is .

Current transformed vertex

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 . 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.

Make a prediction before you reveal the next step.

Decide which controls must change sign before you try it.

Check your reasoning against the live bench.

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?

Use the live bench to test the result before moving on.

Accessibility

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.