Warm up Quadratic Equations in the Real World

Keystrokes:

At the bottom of the display are the coordinates of the maximum point on the graph. The y-value of these coordinates is the maximum value of the function, or 1280.

ENTER2nd [CALC] 4

y = –5x2 + 40x + 1200. What is the maximum monthly income

4-2 Solving Equations by

Graphing

• Root

• Zero

• Solution

• All of these terms mean the x-intercepts of a function, or the x values that make f(x) = 0

Possible solutions

Replace the 0 with a y. So you will have ◦ y=ax2+bx+c

Graph the equation Look at the x-intercepts to find the solution Check your solution

Steps to Solve by Graphing

Example 1

Solve x2 + 6x + 8 = 0 by graphing.

Example 2

Solve x2 – 4x = –4 by graphing.

Example 3 NUMBER THEORY Use a quadratic equation

to find two numbers with a sum of 4 and a product of 5.Understand Let x = one of the numbers. Then

4 – x = the other number.

Solve:Graph the related function.

Example 4

Solve –x2 + 4x – 1 = 0 by graphing.

Use the ZERO feature, (2nd [CALC]), to find the zeros of the function, Use the arrow keys to locate a left bound for the zero and press [ENTER ]. Then locate a right bound and press [ENTER] twice.

Example 5

Locate the zeros in the function 2(x- 3) 2 – 9 = 0

Example 6

ROYAL GORGE BRIDGE The highest bridge in the United States is the Royal Gorge Bridge in Colorado. The deck of the bridge is 1053 feet above the river below. Suppose a marble is dropped over the railing from a height of 3 feet above the bridge deck. How long will it take the marble to reach the surface of the water, assuming there is no air resistance? Use the formula h(t) = –16t

2 + h0, where t is the time in seconds and h0 is the initial height above the water in feet.

Solve by Using a Calculator

Graph the related function y = –16t

2 + 1056 using a graphing calculator. Adjust your window so that the x-intercepts are visible.

Solve 0 = –16t

2 + 1056.