304-Digital Lecture.pptx
- 1. Forms of Quadratics QUADRATIC FUNCTIONS-ALGEBRA 2 CHRISTINE GU
- 2. Lecture Objectives After this lecture, students will be able to: 1. Recognize and graph a quadratic in standard form, vertex form, and intercept form 2. Identify important characteristics of a quadratic from its equation. 3. Transform a quadratic between its three forms by completing the square, multiplying, or factoring California Common Core Standards F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
- 3. What Do You Remember? Recall the parent function of a quadratic function. Discuss with your partner: • What is the vertex of this functions? Axis of symmetry? • Which direction does this parabola open? 𝑓 𝑥 = 𝑥2
- 4. Vertex Form Compared to the parent function 𝑔 𝑥 = 𝑥2, 𝑓 𝑥 has a: • Horizontal shift of h units and a vertical shift of k units • Vertical stretch/compression by a factor of a • Reflection across the x-axis if a < 0 𝑓 𝑥 = 𝑎 𝑥 − ℎ 2 + 𝑘
- 5. Vertex Form (cont.) Vertex: ℎ, 𝑘 Axis of symmetry: 𝑥 = ℎ y-intercept: 0, 𝑓 0 If a > 0, f(x) is concave up. Otherwise if a < 0 , f(x) is concave down. 𝑓 𝑥 = 𝑎 𝑥 − ℎ 2 + 𝑘
- 6. Example Let 𝑓 𝑥 = −2 𝑥 − 1 2 + 3. ℎ = 1, 𝑘 = 3, 𝑎 = −2 vertex: (1, 3) AOS: x = 1 y-intercept: (0, 1) f(x) is concave down 𝑓 0 = −2 0 − 1 2 + 3 = −2 −1 2 + 3 = −2 ∙ 1 + 3 = 1 2. Find the y-intercept: 1. Identify h, k, and a: Graph of f(x) generated with desmos
- 7. Now You Try Let 𝑓 𝑥 = 3 𝑥 − 5 2 − 4. Work with your partner to identify the vertex, axis of symmetry, and y- intercept of this quadratic equation. Then graph f(x) and label the characteristics.
- 8. Vertex Form to Standard Form Expand 𝑓 𝑥 = 𝑎 𝑥 − ℎ 2 + 𝑘 by FOILing and combining like terms. from Friendly Math 101 on Youtube
- 9. Standard Form Vertex: −𝑏 2𝑎 , 𝑓 −𝑏 2𝑎 Axis of symmetry: x = −𝑏 2𝑎 y-intercept: (0, c) If a > 0, f(x) is concave up. Otherwise if a < 0 , f(x) is concave down. 𝑓 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐
- 10. Example Let f x = 3𝑥2 + 6𝑥 + 4. 𝑎 = 3, 𝑏 = 6, 𝑐 = 4 1. Identify a, b, and c: Graph of f(x) generated with desmos 2. Find the vertex: −𝑏 2𝑎 = −6 2 ∙ 3 = −1 𝑓 −1 = 3 −1 2 + 6 −1 + 4 = 3 1 − 6 + 4 = 1 vertex: (-1, 1) AOS: x = -1 y-intercept: (0, 4) f(x) is concave up
- 11. Standard Form to Vertex Form Complete the square for 𝑓 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐. from The Organic Chemistry Tutor on Youtube
- 12. Standard Form and Intercept Form → Factor 𝑓 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐. Example Let 𝑓 𝑥 = −2𝑥2 − 4𝑥 + 6. Work with your partner to f(x) completely. ← multiply/FOIL 𝑓 𝑥 = 𝑎 𝑥 − 𝑝 𝑥 − 𝑞 . 𝑓 𝑥 = −2𝑥2 − 4𝑥 + 6 = − 2 x2 + 2x − 3 = −2(𝑥 + 3)(𝑥 − 1)
- 13. Now You Try Let f x = x − 5 x − 1 . Work with your partner to rewrite f(x) into standard form. Then identify the vertex, axis of symmetry, and y-intercept of this quadratic equation.
- 14. Intercept Form x-intercepts: (p, 0) and (q, 0) vertex: 𝑝+𝑞 2 , 𝑓 𝑝+𝑞 2 Axis of symmetry: x = 𝑝+𝑞 2 y-intercept: 0, 𝑓 0 If a > 0, f(x) is concave up. Otherwise if a < 0 , f(x) is concave down. 𝑓 𝑥 = 𝑎 𝑥 − 𝑝 𝑥 − 𝑞
- 15. Example 1. Identify a, p, and q: Let 𝑓 𝑥 = −2(𝑥 + 3)(𝑥 − 1) a = −2, p = −3, q = 1 2. Find the vertex: 𝑝 + 𝑞 2 = −3 + 1 2 = −1 3. Find the y-intercept: 𝑓 0 = −2 0 + 3 0 − 1 = −2 3 −1 = 6 x-ints: (-3, 0) and (1, 0) vertex: (-1, 8) AOS: x = -1 y-intercept: (0, 6) f(x) is concave down f −1 = −2 −1 + 3 −1 − 1 = −2 2 −2 = 8 Graph of f(x) generated with desmos
- 16. Summary Vertex Form 𝑓 𝑥 = 𝑎 𝑥 − ℎ 2 + 𝑘 • v: (h, k) Standard Form 𝑓 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 • y-intercept: (0, c) • v: −𝑏 2𝑎 , 𝑓 −𝑏 2𝑎 Intercept Form 𝑓 𝑥 = 𝑎 𝑥 − 𝑝 𝑥 − 𝑞 • x-intercepts: (p, 0), (q, 0) • v: 𝑝+𝑞 2 , 𝑓 𝑝+𝑞 2 Expand Factor Complete the square Multiply/FOIL
- 17. Now You Try Let 𝑓 𝑥 = −𝑥2 + 4𝑥 + 5. a. What form is this? Rewrite f(x) into the other two forms. b. Identify the vertex, axis of symmetry, x- and y-intercepts, and concavity. c. Graph and label f(x).