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Characteristics of Quadratic Functions

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Page 1: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that

Characteristics of Quadratic Functions

Page 2: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that

Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x-value that makes the function equal to 0. So a zero of a function is the same as an x-intercept of a function. Since a graph intersects the x-axis at the point or points containing an x-intercept, these intersections are also at the zeros of the function. A quadratic function may have one, two, or no zeros.

Zeroes of a Function

Page 3: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that

Find the zeros of the quadratic function from its graph. Check your answer.

y = x2 – 2x – 3

The zeros appear to be (–1, 0) and (3, 0).

y = (–1)2 – 2(–1) – 3 = 1 + 2 – 3 = 0

y = 32 –2(3) – 3 = 9 – 6 – 3 = 0

y = x2 – 2x – 3

Check

Page 4: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that

A vertical line that divides a parabola into two symmetrical halves is the axis of symmetry. The axis of symmetry always passes through the vertex of the parabola. You can use the zeros to find the axis of symmetry.

Axis of Symmetry

Page 5: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that
Page 6: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that

Find the axis of symmetry of each parabola.

A. (–1, 0) Identify the x-coordinate of the vertex.

The axis of symmetry is x = –1.

Find the average of the zeros.

The axis of symmetry is x = 2.5.

B.

Page 7: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that

If a function has no zeros or they are difficult to identify from a graph, you can use a formula to find the axis of symmetry. The formula works for all quadratic functions.

Page 8: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that

Find the axis of symmetry of the graph of y = –3x2 + 10x + 9.

Step 1. Find the values of a and b.

y = –3x2 + 10x + 9

a = –3, b = 10

Step 2. Use the formula.

The axis of symmetry is

Page 9: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that

Find the axis of symmetry of the graph of y = 2x2 + x + 3.

Step 1. Find the values of a and b.

y = 2x2 + 1x + 3a = 2, b = 1

Step 2. Use the formula.

The axis of symmetry is .

Page 10: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that

Once you have found the axis of symmetry, you can use it to identify the vertex.

Page 11: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that

Find the vertex.

y = 0.25x2 + 2x + 3

Step 1 Find the x-coordinate of the vertex. The zeros are –6 and –2.

Step 2 Find the corresponding y-coordinate.y = 0.25x2 + 2x + 3

= 0.25(–4)2 + 2(–4) + 3 = –1 Step 3 Write the ordered pair.

(–4, –1)

Use the function rule.

Substitute –4 for x .

The vertex is (–4, –1).

Page 12: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that

The graph of f(x) = –0.06x2 + 0.6x + 10.26 can be used to model the height in meters of an arch support for a bridge, where the x-axis represents the water level and x represents the distance in meters from where the arch support enters the water. Can a sailboat that is 14 meters tall pass under the bridge? Explain.

The vertex represents the highest point of the arch support. The vertex is at (5, 11.76). It is only 11.76 feet high, so it won’t fit.

Page 13: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that

The height of a small rise in a roller coaster track is modeled by f(x) = –0.07x2 + 0.42x + 6.37, where x is the distance in feet from a supported pole at ground level. Find the height of the rise.

Step 1 Find the x-coordinate.

a = – 0.07, b= 0.42 Identify a and b.

Substitute –0.07 for a and 0.42 for b.

Replace x with 3-0.07(3)2 + 0.42(3) + 6.37

The rise is 7 feet high.

Page 14: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that

1. Find the zeros and the axis of symmetry of the parabola.

2. Find the axis of symmetry and the vertex of the graph of y = 3x2 + 12x + 8.

zeros: (–6, 0), (2, 0); Axis of symmetry: x = –2

x = –2; (–2, –4)

Try these…

Page 15: Characteristics of Quadratic Functions. Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x- value that

25 feet

3. The graph of f(x) = –0.01x2 + x can be used to model the height in feet of a curved arch support for a bridge, where the x-axis represents the water level and x represents the distance in feet from where the arch support enters the water. Find the height of the highest point of the bridge.


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