QUADRATIC FUNCTIONS

IN THE QUADRATIC FUNCTION Y = AX 2 + BX + C…  What does the “a” tell you?  The width of the parabola  The greater the |a| the more narrow the parabola  The smaller the |a| the wider the parabola  The way the parabola opens  When a > 0, the parabola opens “up”  When a < 0, the parabola opens “down”

IN THE QUADRATIC FUNCTION Y = AX 2 + BX + C…  What does “c” tell you?  This is the y-intercept  When b = 0, it is also the vertex.

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 = x 2 – 2x – 3 The zeros appear to be (–1, 0) and (3, 0). y = (–1) 2 – 2(–1) – 3 = – 3 = 0 y = 3 2 –2(3) – 3 = 9 – 6 – 3 = 0 y = x 2 – 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 = – 3x x + 9. Step 1. Find the values of a and b. y = –3x x + 9 a = –3, b = 10 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 = 0.25x 2 + 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 = 0.25x 2 + 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).

TO GRAPH A QUADRATIC FUNCTION…  Find the axis of symmetry (using x = -b/2a)  Find the vertex (by plugging in the value of “x” to find “y”.  Find the y-intercept (“c”).  Graph the reflection of “c” across the axis of symmetry.

GRAPH Y = X 2 – 4X - 5  a = 1, b = -4 and c = -5  Find the axis of symmetry. x = -b/2a----x = -(-4)/2(1)  x = 2  Replace x with 2. y = (2) 2 – 4(2) – 5  y = 4 – 8 – 5  y = -9  (2, -9)

GRAPH Y = X 2 – 4X – 5 CONTINUED  The y-intercept is (0, -5).  It is 2 away from the axis of symmetry. So there is another point that is 2 away on the other side of the axis of symmetry.

The graph of f(x) = –0.06x x 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 feet high, so it won’t fit.

The height of a small rise in a roller coaster track is modeled by f(x) = – 0.07x x , 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) (3) The rise is 7 feet high.

Another Example The height in feet of a basketball that is thrown can be modeled by f(x) = –16x x, where x is the time in seconds after it is thrown. Find the basketball’s maximum height and the time it takes the basketball to reach this height. Then find how long the basketball is in the air. In one second, the basketball will be 16 feet above the ground. It is in the air for 2 seconds.

Yet Another Example As Molly dives into her pool, her height in feet above the water can be modeled by the function f(x) = –16x x, where x is the time in seconds after she begins diving. Find the maximum height of her dive and the time it takes Molly to reach this height. Then find how long it takes her to reach the pool. In ¾ of a second Molly will be 9 feet above the water. She is out of the water 1.5 seconds.