Graphing Quadratic Functions in Vertex Form: Lesson Plan

In vertex form, y = a(x − h)² + k, a quadratic tells you its turning point directly: the vertex is at (h, k). The number a says whether the parabola opens up or down and how narrow it is. Three sliders, one per letter, make each role obvious.

Learning objectives

• Read the vertex, the axis of symmetry and the direction of opening from vertex form.
• Describe the effect of a, h and k on the graph.
• Convert standard form to vertex form by completing the square.

The opening question

Show y = x² and y = (x − 2)² − 3 on the same axes: "What did the −2 and the −3 do?" Students often expect (x − 2) to move the graph left. It moves right, and explaining why is the heart of the lesson.

The lesson, click by click

1. Click 0: y = x², vertex at the origin.
2. Click 1, slider k: the parabola moves up and down; the vertex is at height k.
3. Click 2, slider h: it moves sideways. (x − 2)² is zero at x = 2, so the lowest point is at x = 2: a right shift.
4. Click 3, slider a: bigger |a| is narrower; negative a flips it upside down, and the vertex becomes a maximum.
5. Click 4: from standard form, y = x² − 4x + 1 = (x − 2)² − 4 + 1 = (x − 2)² − 3.

Common mistakes to watch for

• The sign of h. In (x + 3)², h = −3, so the shift is left.
• Forgetting to subtract when completing the square. x² − 4x = (x − 2)² − 4, not (x − 2)².
• a outside the square. 2(x − 1)² is not (2x − 2)², which equals 4(x − 1)².
• Vertex versus intercept. The y-intercept is the value at x = 0 (here 1), not k.

Check questions

1. Vertex and direction of y = −2(x + 1)² + 5? ((−1, 5), opens down, maximum value 5.)
2. Write y = x² + 6x + 2 in vertex form. ((x + 3)² − 7.)
3. Where does y = (x − 2)² − 3 cross the x-axis? (x = 2 ± √3, about 0.27 and 3.73.)

Building it in Tracé

1. Type y = a (x - h)^2 + k and press Plot this equation as a graph.
2. Tick Live slider for a, h and k, and Mark the maxima and minima so the vertex is labelled as it moves.
3. Show the sliders one per click, in the order k, h, a.

Questions

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