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: |x 2 – 4x – 3| = x 3 + 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) = x 3 – 8 ◦ The zeros of the function f are the solutions (or roots) of the equation.  Example 2: x 4 – 2x 2 -3x – 2  See graphing calculator  Example 3: x 5 – x 3 + x 2 – 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 x 4 + x 2 – 2x – 1 = 0 and find solutions

 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 2x 2 + x – 1 = 0 ◦ Plug answers into the denominator and discard any value that would also make the denominator equal 0.

 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)