Essential Question: What are the zeroes of a function?

Solution: A number that, when substituted, produces a true statement.◦ 3x + 2 = 17 → 3(5) + 2 = 17◦ 5 is a solution

Solve: To solve an equation means to find all of its solutions

Two equations are equivalent if they have the same solutions.◦ 3x + 2 = 17 and x – 2 = 3 are equivalent because

5 is the only solution to both

Example 1: The Intersection Method◦ Graph both equations on the same screen◦ Find the x-coordinate of each point of intersection◦ Ex: |x2 – 4x – 3| = x3 + x – 6

See graphing calculator Graph → more → Math → more → ISECT

The x-intercept method◦ A zero of a function is an input (x-value) that

produces an output of 0. E.g. 2 is a zero of the function f(x) = x3 – 8

◦ The zeros of the function f are the solutions (or roots) of the equation. Example 2: x4 – 2x2 -3x – 2

See graphing calculator Example 3: x5 – x3 + x2 – 5 = 0

Using ‘Root’ to solve In graph menu, ‘More’, ‘Math’, ‘Root’

Example 3: Technology quirk #1◦ Square root functions:

◦ Square root = 0 when function = 0 So we can graph x4 + x2 – 2x – 1 = 0 and find

solutions

4 2 2 1 0x x x

Technology quirk #2 Solving f (x) / g (x) = 0

◦ Solve

◦ Almost impossible to read using the graphing calculator, but rules of fractions tell us that a fraction = 0 only when the numerator = 0. So, all we truly have to graph is 2x2 + x – 1 = 0

◦ Plug answers into the denominator and discard any value that would also make the denominator equal 0.

2

2

2 10

9 9 2

x x

x x

Summary◦ Intersection Method

Graph both lines Find the x-coordinate of each point of intersection

◦ x-Intercept Method Rewrite the function as f (x) = 0 Graph y = f (x) Find the x-intercepts of the graph of f (x). Those x-

intercepts are the solutions of the equation. Assignment

◦ Pg. 87, 1-43 (odd problems)