Chapter 3: Functions and Graphs 3.3: Quadratic Functions

1. Chapter 3: Functions and Graphs3.3: Quadratic Functions Essential Question(s):How can you tell if a quadratic function opens up or down has a minimum or maximum, and how many x-intercepts it has?

2. 3.3: Quadratic Functions • “Wait… didn’t we do this already?” • I tried to warn you… • The notes that follow in yellow, I will expect you to memorize (meaning: they won’t be given to you on a quiz) • Quadratic Functions are parabolas (‘U’ shaped) and • Can open either upward or downward • Always have a vertex which is either the maximum or minimum • Opening up == minimum, opening down == maximum • Always have exactly one y-intercept • Can have 0, 1, or 2 x-intercepts • The x-intercept(s) are the solution(s) [roots] of the equation

3. 3.3: Quadratic Functions • Quadratic Functions can be written in one of three forms • Transformation form: f (x) = a(x – h)2 + k • Most useful for finding the vertex of a parabola • Vertex is at (h, k) • (Set inside parenthesis = 0 & solve, number outside) • If a is positive, the graph opens up. • If a is negative, graph opens down. • The y-intercept is at ah2 + k • The x-intercepts are at

4. 3.3: Quadratic Functions h = 2 and k = -5, so vertex is at (2, -5) Because a = -6, graph opens down There is no h, and k = 1 so vertex is at (0, 1) Because a = -1, graph opens down • Using Transformation Form • Find the vertex of the function and state whether the graph opens upward or downward • g(x) = -6(x – 2)2 – 5 • h(x) = -x2 + 1

5. 3.3: Quadratic Functions • Polynomial form: f (x) = ax2 + bx + c • Yeah, we’ve seen this plenty already… • Most useful for finding the y-intercept • y-intercept is at (0, c) • If a is positive, the graph opens up. • If a is negative, graph opens down. • The vertex is at • The x-intercepts are at

6. 3.3: Quadratic Functions The y-intercept is at (0, -1) Because a = 1, graph opens up The y-intercept is at (0, 5) Because a = 2, graph opens up • Using Polynomial Form • Determine the y-intercept and state whether the graph opens upward or downward • g(x) = x2 + 8x – 1 • g(x) = 2x2 – x + 5

7. 3.3: Quadratic Functions • x-intercept form: f (x) = a(x – s)(x – t) • This is simply polynomial form factored out • Most useful for finding the x-intercepts (duh) • x-intercepts are at (s, 0) and (t, 0) • If a is positive, the graph opens up. • If a is negative, graph opens down. • The vertex is at • The y-intercepts is at (0, ast)

8. 3.3: Quadratic Functions The x-intercept are at (-3, 0) and (-1, 0) Because a = -2, graph opens down The x-intercepts are at (-2.1, 0) and (0.7, 0) Because a = -0.4, graph opens down • Using x-intercept Form • Determine the x-intercepts and state whether the graph opens upward or downward • h(x) = -2(x + 3)(x + 1) • f(x) = -0.4(x + 2.1)(x – 0.7)

9. 3.3: Quadratic Functions • Assignment • Page 170 • Problems 1-25, odd problems