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QUADRATIC FUNCTION

A function of the form

where a, b, and c, are real numbers with a ≠ 0, is called a quadratic function.

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THE STANDARD FORM OFA QUADRATIC FUNCTION

The quadratic function

is in standard form. The graph of f is a parabola with vertex (h, k). The parabola is symmetric with respect to the line x = h, called the axis of symmetry of the parabola. If a > 0, the parabola opens up and k is the minimum value of f, and if a < 0, the parabola opens down and k is the maximum value of f.

Vertex

The lowest or highest point of a parabola.

Vertex

Axis of symmetry

The vertical line through the vertex of the parabola.

Axis of

Symmetry

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EXAMPLE 1 Finding a Quadratic Function

Find the standard form of the quadratic function whose graph has vertex (–3, 4) and passes through the point (–4, 7).

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PROCEDURE FOR GRAPHING f (x) = a(x – h)2 + k

Step 1 The graph is a parabola. Identify a, h, and k.

Step 2 Determine how the parabola opens. If a > 0, the parabola opens up. If a < 0, the parabola opens down.

Step 3 Find the vertex. The vertex is (h, k). If a > 0 (or a < 0), the function f has a minimum (or a maximum) value k at x = h.

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PROCEDURE FOR GRAPHING f (x) = a(x – h)2 + k

Step 4 Find the x-intercepts (if any). Set f (x) = 0 and solving the equation a(x – h)2 + k = 0 for x.

If the solutions are real numbers, they are the x-intercepts. If not, the parabola either lies above the x-axis (when a > 0) or below the x-axis (when a < 0).

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PROCEDURE FOR GRAPHING f (x) = a(x – h)2 + k

Step 6 Sketch the graph. Plot the points found in Steps 3–5 and join them by a parabola. Show the axis x = h of the parabola by drawing a dashed line.

Step 5 Find the y-intercept. Replace x with 0. Then f (0) = ah2 + k is the y-intercept.

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EXAMPLE 2Graphing a Quadratic Function in Standard Form

Sketch the graph of 23 2 12.f x x

SolutionStep 1 a = –3, h = –2, and k = 12

Step 2 a = –3, a < 0, the parabola opens down.

Step 3 (h, k) = (–2, 12); the function f has a maximum value 12 at x = –2.

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EXAMPLE 2Graphing a Quadratic Function in Standard Form

Step 4 Set f (x) = 0 and solve for x.

2

2

2

0 3 2 12

12 3 2

4 2

x

x

x

2 2

0 or 4

-intercepts: 0 and 4

x

x x

x

Solution continued

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EXAMPLE 2Graphing a Quadratic Function in Standard Form

Solution continued

Step 5 Replace x with 0.

20 3 0 2 12

3 4 12 0

-intercept is 0 .

f

y

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EXAMPLE 2Graphing a Quadratic Function in Standard Form

Solution continued

Step 6 axis: x = –2

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PROCEDURE FOR GRAPHING f (x) = ax2 + bx + c

Step 1 The graph is a parabola. Identify a, b, and c.

Step 2 Determine how the parabola opens. If a > 0, the parabola opens up. If a < 0, the parabola opens down.

Step 3 Find the vertex (h, k). Use the formula:

, , .2 2

b bh k f

a a

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Step 4 Find the x-intercepts (if any). Let y = f (x) = 0. Find x by solving the equation ax2 + bx + c = 0. If the solutions are real numbers, they are the x-intercepts. If not, the parabola either lies above the x-axis (when a > 0) or below the x-axis (when a < 0).

PROCEDURE FOR GRAPHING f (x) = ax2 + bx + c

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Step 5 Find the y-intercept. Let x = 0. The result f (0) = c is the y-intercept.

Step 7 Draw a parabola through the points found in Steps 3–6.

Step 6 The parabola is symmetric with

respect to its axis,

Use this symmetry to find additional

points.

.2

bx

a

PROCEDURE FOR GRAPHING f (x) = ax2 + bx + c

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EXAMPLE 3Graphing a Quadratic Function f (x) = ax2 + bx + c

Solution

Sketch the graph of 22 8 10.f x x x

Step 1 a = 2, b = 8, and c = –10

Step 2 a = 2, a > 0, the parabola opens up.

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EXAMPLE 3Graphing a Quadratic Function f (x) = ax2 + bx + c

Step 3 Find (h, k).

2

22 2

2 2 2 8 2 10 18

, 2, 18

8

2

bh

a

k f

h k

Minimum value of –18 at x = –2

Solution continued

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EXAMPLE 3Graphing a Quadratic Function f (x) = ax2 + bx + c

Solution continued

Step 4 Let f (x) = 0.

22 8 10 0x x

x-intercepts: –5 and 1

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EXAMPLE 3Graphing a Quadratic Function f (x) = ax2 + bx + c

Solution continued

20 2 0 8 0 10

-intercept is 10 .

f

y

Step 5 Let x = 0.

Step 6 Axis of symmetry is x = –2. The symmetric image of (0, –10) with respect to the axis x = –2 is (–4, –10).

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EXAMPLE 3Graphing a Quadratic Function f (x) = ax2 + bx + c

Solution continued

Step 7 Sketch the parabola passing through the points found in Steps 3–6.

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EXAMPLE 4Identifying the Characteristics of a Quadratic Function from Its Graph

The graph of f (x) = –2x2 +8x – 5 is shown.Find the domain and range of f.

Solution

The domain is (–∞, ∞). The range is (–∞, 3].