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Published byAmberlynn Lucas Modified over 9 years ago
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Essential Question: What are the zeroes of a function?
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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
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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
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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’
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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
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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.
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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)
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