1. M3U6D6 Warm-up: Find the zeroes: 1.x 2 – 6x – 16 = 0 2.2x 2 + 7x + 5 = 0 (x-8)(x+2)=0 x-8=0 or x+2=0 x=8 or x=-2 (2x+5)(x+1)=0 2x+5=0 or x+1=0 2x=-5 or x=-1 x=-5/2

2. HW Check: Document camera

3. M1U6D6 Axis of Symmetry and Graphing Quadratics Objective: To explore and graph quadratic functions.

4. Quadratic functions of the forms y = ax 2 and y = ax 2 + c DEFINITIONS…. Standard Form of a Quadratic Function: A quadratic function is a function that can be written in the form y = ax 2 + bx + c, where a ≠ 0. This form is called the STANDARD FORM OF A QUADRATIC FUNCTION. Parabola: a “U-Shaped curve” Axis of symmetry: The fold or line that divides the parabola into two matching halves. Exploring Quadratic Graphs

5. Vertex: If a > 0 inIf a < 0 in y = ax 2 + bx + c then vertex is a minimumvertex is a maximum Exploring Quadratic Graphs

6. Identify the vertex of each graph. Tell whether it is a minimum or a maximum. Exploring Quadratic Graphs a. The vertex is (1, 2). b. The vertex is (2, –4). It is a maximum.It is a minimum.

7. Use the graphs below. Order the quadratic functions  (x) = –x 2,  (x) = –3x 2, and  (x) = x 2 from widest to narrowest graph. So, the order from widest to narrowest is  (x) = x 2,  (x) = –x 2,  (x) = –3x 2. 1212 1212  (x) = –x 2  (x) = x 2 1212 Of the three graphs,  (x) = x 2 is the widest and  (x) = –3x 2 is the narrowest. 1212  (x) = –3x 2 Exploring Quadratic Graphs

8. Reminder….. +a in y = ax 2 + bx + c makes it open up. -a in y = ax 2 + bx + c makes it open down. If |a| is “small” (i.e. between 0 and 1) the graph is wide. If |a| is “big” (i.e. greater than 1) the graph is narrow.

9. PROPERTY: Graph of a Quadratic Function: The graph of y = ax 2 + bx + c, where a ≠ 0, has the line x = -b/2a as its axis of symmetry. The x-coordinate of the vertex is –b/2a.

10. Graph the function y = 2x 2 + 4x – 3. Step 1: Find the equation of the axis of symmetry and the coordinates of the vertex. Find the equation of the axis of symmetry.x = b2ab2a – = –4 2(2) = – 1 The x-coordinate of the vertex is –1. y = 2x 2 + 4x – 3 y = 2(–1) 2 + 4(–1) – 3 = –5 To find the y-coordinate of the vertex, substitute –1 for x. The vertex is (–1, –5).

11. Step 2: Find two other points. Use the y-intercept. For x = 0, y = –3, so one point is (0, –3). Choose a value for x on the same side of the vertex. Let x = 1 y = 2(1) 2 + 4(1) – 3 = 3 For x = 1, y = 3, so another point is (1, 3). Find the y-coordinate for x = 1.

12. Step 3: Reflect (0, –3) and (1, 3) across the axis of symmetry to get two more points. Then draw the parabola.

13. Aerial fireworks carry “stars,” which are made of a sparkler-like material, upward, ignite them, and project them into the air in fireworks displays. Suppose a particular star is projected from an aerial firework at a starting height of 610 ft with an initial upward velocity of 88 ft/s. How long will it take for the star to reach its maximum height? How far above the ground will it be? The equation h = –16t 2 + 88t + 610 gives the height of the star h in feet at time t in seconds.

14. Step 2: Find the h-coordinate of the vertex. h = –16(2.75) 2 + 88(2.75) + 610Substitute 2.75 for t. h = 731Simplify using a calculator. The maximum height of the star will be about 731 ft. Step 1: Find the x-coordinate of the vertex. After 2.75 seconds, the star will be at its greatest height. b2ab2a – = –88 2(–16) = 2.75

15. Classwork: M1U6D6 CW Homework: M1U6D6 HW