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Grade 9 · Quadratic functions

Robot controller: Connect symmetry and quadratic values

Robot controller investigation: connect symmetry and quadratic values. Change the quantities, follow the motion, and explain the result using the displayed mathematical relationship.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Robot controller: Connect symmetry and quadratic values. Moving input: -4. Output: 0. Vertex: (-2, -2). Value at x=2: 6y = 0.5(x−(-2))²+(-2)-8-4048-40040xRead numeric axes; drawing scales differ.
Inputs are real and the fixed graph window is x∈[−8,8], y∈[−40,40]; curves are clipped outside it and axes have different drawing scales. A zero leading coefficient is explicitly treated as a constant function.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Moving input
-4
Output
0
Vertex
(-2, -2)
Value at x=2
6

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

Vertex form separates opening and curvature from horizontal and vertical shifts. Equal offsets on either side of the symmetry line give equal outputs. When the leading coefficient is zero, the rule is constant rather than quadratic, and it has no unique vertex. This investigation begins with leading coefficient a = 0.5; horizontal shift h = -2; vertical shift k = -2. Treat the named setting as a constructed classroom model, then explain which relationships would still hold if its numbers changed.

A relationship to keep

y = a(x−h)²+k

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Represent the given quantities

    Locate (h,k) when a is nonzero and read the symmetry line x=h.

  2. STEP 2

    Follow the changing model

    Playback moves x from −4 to 4 and recalculates y, while the graph shows the selected transformation.

  3. STEP 3

    Check and explain the relationship

    Compare equal offsets from h. Use the sign of a to identify a minimum or maximum, with a=0 handled separately.

Your turn to explain

Make a prediction. Test your reasoning.

Robot controller: For y=0.5(x−(-2))²+(-2), find y at x=2.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

6. Evaluate the entire difference before squaring, then multiply and shift.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 9 · Algebra

Test whether a table has constant rate

Compare output changes over equal input intervals.