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ScientistGrades 9–10⏱ 20 min

Quadratic functions

Drag a parabola’s vertex and roots. Watch the same curve in vertex, standard and factored form.

Standards: HSF-IF.C.7aHSF-IF.C.8aHSA-SSE.B.3

1Wonder

A basketball shot, the arch of a bridge, the water from a fountain: they all follow the same curve. What is it, and how can one equation tell us where it peaks and where it lands?

2Explore

Parabola lab

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Turn your phone sideways, or open this lesson on a tablet or computer. The explanation and quiz below work here.

−8−6−4−22468−6−4−2246Vertex formy = (x − 1)² − 4Standard formy = x² − 2x − 3Factored formy = (x + 1)(x − 3)Vertex (1, −4) · minimumRoots: x = −1, x = 3y-intercept: (0, −3)b² − 4ac = 16 → 2 rootsSillyChili
Change it with
Try this
  • Drag the purple vertex. Watch all three equations change together.
  • Change a to a negative number. What happens to the parabola? Is the vertex now a maximum or a minimum?
  • Drag the red roots apart, then together until they meet. What does b² − 4ac say?
  • Move the vertex above the x-axis with the parabola opening up. Can you still use Factored form?

3Explain

−39−24−1100112439xx²y = x²SillyChili
Step 1 of 6

The parabola y = x²

Square each x and plot the points. They make a U-shaped curve called a parabola. It is symmetric: x = 2 and x = −2 give the same y.

Three forms of a quadraticy=a(x−h)2+ky=ax2+bx+cy=a(x−r)(x−s)\begin{aligned} y &= a(x-h)^2 + k \\ y &= ax^2 + bx + c \\ y &= a(x-r)(x-s) \end{aligned}

Vertex form shows the vertex (h, k), standard form the y-intercept c, and factored form the roots r and s.

4Practice

Question 1 of 5

What is the vertex of y = (x − 3)² + 2?

5Recap

  • The graph of a quadratic is a parabola. It opens up when a > 0 and down when a < 0.
  • Vertex form a(x − h)² + k: the vertex is (h, k), the axis of symmetry is x = h.
  • Factored form a(x − r)(x − s): the roots are r and s.
  • Standard form ax² + bx + c: the y-intercept is c; b² − 4ac counts the roots.