characteristics of quadratic functions. recall that an x-intercept of a function is a value of x...
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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 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
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
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
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.
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.
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
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 .
Once you have found the axis of symmetry, you can use it to identify the vertex.
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).
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.
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.
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…
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.