Math With AmarA C A D E M Y
All math animations

High school · Functions

Move and reshape a parabola

Control opening, horizontal shift, and vertical shift while a point traces the graph.

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

Watch the relationship

Paused
Move and reshape a parabola. Vertex: (0, -1). Moving x: -2. Moving y: 2. Shape: Opens upwardy = a(x − h)² + k-4-2024-606xTeal: vertex · amber: tracing point
The fixed viewing window is x from −5 to 5 and y from −6 to 6; the two axes have different drawing scales. Curve and moving point portions outside this window are clipped. At a = 0 there is no unique vertex.

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.

Vertex
(0, -1)
Moving x
-2
Moving y
2
Shape
Opens upward

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 three different transformations. The parameter h moves the symmetry line to x = h, k moves the graph vertically, and a controls curvature and opening direction. When a = 0 the quadratic term disappears, leaving a constant function rather than a parabola.

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

    Locate the turning point

    For nonzero a, the teal vertex is at (h, k) and the dashed symmetry line is x = h. Substituting x = h makes the squared term zero.

  2. STEP 2

    Change the opening

    A positive a opens upward and a negative a opens downward. A larger absolute value changes y more for the same distance from the symmetry line. At a = 0, the graph becomes the horizontal line y = k.

  3. STEP 3

    Trace equal offsets

    The amber point moves from h − 2 to h + 2. Compare positions the same distance to either side of h; squaring the opposite offsets produces the same height.

Your turn to explain

Make a prediction. Test your reasoning.

For y = −2(x − 1)² + 3, identify the vertex and calculate y at x = 0 and x = 2.

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

Compare your explanation

The vertex is (1, 3), the parabola opens downward, and y = 1 at both x = 0 and x = 2. These points are symmetric about x = 1.

Work through a full lesson

Connect the animation to a worked example and practice questions.

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