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1. Parabolas by Dr. Carol A. Marinas

2. Transformation Shifts Tell me the following information about: f(x) = (x – 4)2 – 3 • What shape is the graph? • What direction is it going? • What is the vertex? • Is it a high point or low point? • What is the y-intercept?

3. Transformation Shifts Tell me the following information about: f(x) = (x – 4)2 – 3 • What shape is the graph? parabola • What direction is it going? up • What is the vertex? (4, –3) • Is it a high point or low point? Low point • What is the y-intercept? (0, 13)

4. y =(x – 4)2 – 3

5. Standard Form • y = (x – h)2 + k • Vertex is (h, k) • Line of Symmetry is x = h

6. Standard to General Form • y =(x – 4)2 – 3 • y = (x2 – 8x + 16) – 3 • y = x2 – 8x + 13

7. y = x2 – 8x + 13 y – 13 = x2 – 8x y – 13 + 16 = x2 – 8x + 16 y + 3 = (x – 4)2 y = (x – 4)2 – 3 Vertex is (4, – 3) Get ‘a’ equal to 1 by multiplying or dividing the equation. (done) Move constant to left side Complete the square Solve for y General to Standard Form

8. General Form • y = ax2 + bx + c Vertex is ( –b/2a, f(–b/2a) ) y-intercept is (0, c) Line of Symmetry is x = –b/2a • Example: y = x2 – 8x + 13 Vertex is (–(– 8)/2(1), f (8/2)) or (4, f(4)) or (4, –3) y-intercept is (0, 13) Line of Symmetry is x = 4

9. x-intercepts • For x-intercepts, the y value is 0. • y = ax2 + bx + c becomes • 0 = ax2 + bx + c which is a quadratic equation that is solved by factoring or the quadratic formula.

10. x-intercepts usingQuadratic Formula • 0 = ax2 + bx + c • x = • b2 – 4ac is the discriminant and is used to tell us how many x-intercepts exist.

11. Discriminant if less than 0, no x-intercepts b2 – 4ac if it is 0, 1 x-intercept if greater than 0, 2 x-intercepts Example: y = x2 – 8x + 13 The discriminant is (–8)2 – 4(1)(13) or 64 – 52 or 12 Since 12 > 0, there are 2 x-intercepts.

12. Finding the x-intercepts Ex: y = x2 – 8x + 13 The discriminant is 12. • To actually find the x-intercepts, let’s continue using the Quadratic Formula. • x = = • x = 8 ±√12 = 8 ± 2√3 = 4 ± 2 2 The x-intercepts are (4 – , 0 ) and (4 + , 0)

13. Final Graph of y = (x – 4)2 –3 or y = x2 – 8x + 13

14. Review • Standard Form y = (x – h)2 + k • General Form y = ax2 + bx + c Know how to find the following: * Vertex * y-intercept(s) * High/Low Point * x-intercept(s) * Axis of Symmetry * Graph the parabola