S. D. V. Stephens

notes / Integrated Mathematics / chapter 9: Quadratics

9.6 Graphs of Quadratics

The graph of a quadratic is a parabola. Its direction, intercepts and vertex can all be read off the equation, and completing the square puts the equation in a form that shows the vertex directly.

For : if the parabola opens up, and if it opens down.

Two parabolas: the one labelled a>0 opens upward and the one labelled a<0 opens downward.
  • -intercept: .
  • -intercepts: solve .
  • Vertex: , or complete the square for vertex form.

Example 9.6.1. Graph .

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  1. Vertex: , since and .
  2. -intercept: , since .
  3. -intercepts: solve : so and .

Vertex form, by completing the square (X: top , sides and , bottom ): Reading it off: the in front means the parabola opens up, the makes it narrow, shifts it left 2, and shifts it down 3.

The -intercepts from vertex form:

Graph of f(x)=2x^2+8x+5, an upward parabola with vertex (-2,-3), y-intercept (0,5) and x-intercepts near -3.22 and -0.78.

Example 9.6.2. Graph .

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  1. Vertex: . Using : Completing the square instead (X: top , sides and , bottom ): so the graph is shifted left and up .

  2. -intercept: .

  3. -intercepts: and . For the X (top , bottom ) does not factor nicely, so use the formula: Checking: and . From vertex form:

Graph of g(x)=-2x^2-4x+2, a downward parabola with vertex (-1,4), y-intercept (0,2) and x-intercepts near -2.41 and 0.41.

Example 9.6.3. Graph .

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For the vertex: , and

  1. Vertex: .
  2. -intercept: .
  3. -intercept: (the vertex sits on the -axis).
Graph of f(x)=-4x^2-4x-1, a narrow downward parabola whose vertex (-1/2,0) touches the x-axis, passing through (0,-1) and (-1,-1).

Example 9.6.4. Graph .

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  • Vertex: , so the vertex is .
  • -intercepts: none.
  • -intercept: .
Graph of g(x)=3x^2+2x+5, an upward parabola with vertex (-1/3,14/3) and y-intercept (0,5) that never meets the x-axis.

Finding the equation of a parabola

Example 9.6.5. Find a parabola with a vertex of and a -intercept of .

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Find the function in the form . We know the -intercept is , , and .

The vertex gives (with this has ). To get the right -intercept, put in : So

Exercises

Exercise 9.6.6. Let be a quadratic polynomial such that , , . Find .

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, so . Completing the square, . By Vieta, and . From : , and also where , so which only gives back .

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We know , , , and want . for some .

. We also know ; together, Now since .

(Overthinking it.)

From , , so and .

Check: , and :)

So , and at : .