1. Composition of Quadratic Equations 21 October 2010

2. Composition Checklist In order to graph a quadratic equation, you need to know: 1. Roots (x-intercepts) 2. Classify the Roots 3. Concave Up or Down 4. Vertex Point 5. y-intercept

3. Concavity Describes the way a curve bends Concave up means that the graph opens up Concave up means that the graph opens down

4. Concavity, cont. Concave UPConcave DOWN

5. Example -x 2 = -x – 6

6. Your Turn: With your partner, graph the equations on “The Composition of Quadratic Equations” handout in your graphing calculator. Do not sketch a graph of the equation yet!!! Describe each equation as either concave up or concave down.

7. Observations? In standard quadratic form, If the leading coefficient is positive, then the graph is concave up. If the leading coefficient is negative, then the graph is concave down.

8. Vertex of a Parabola The vertex is a point Translation: It has x and y coordinates It is either the minimum or the maximum point of the parabola Depends on the concavity Concave up = vertex is the minimum pt Concave down = vertex is the maximum pt

9. Vertex, cont. Concave UPConcave DOWN Vertex (Minimum) Vertex (Maximum)

10. Vertex Point Formula

11. Solving for the Vertex Point Step 1: Convert the equation into standard quadratic form: y = ax 2 + bx + c Step 2: Use to solve for the x-coordinate of the vertex point You’re only half done!

12. Solving for the Vertex Point, cont. Step 3: Substitute the value of the x- coordinate into the quadratic equation. Step 4: Solve for y. Step 5: Write the x-coordinate and the y-coordinate as a point.

13. Example -x 2 = -x – 6

14. Your Turn: With your partner, solve for the vertex point of the quadratic equations on “The Composition of Quadratic Equations” handout.

15. y-intercept In standard quadratic form, the y- intercept is the c value. y = ax 2 + bx + c y-intercept

16. Example -x 2 = -x – 6

17. Your Turn: With your partner, identify the y- intercept of the quadratic equations on “The Composition of Quadratic Equations” handout.

18. Sketching Graphs of Quadratic Equations We put together all the information that we’ve gathered in the previous steps. Step 1: Graph the roots, vertex point, and y-intercept Step 2: Remind yourself if the graph is concave up or concave down. Step 3: Sketch the graph of the equation accordingly.

19. Example -x 2 = -x – 6 or y = x 2 – x – 6 Roots: x = -3, 2 Vertex Point: (.5, -6.25) y-intercept: y-int. = -6 Concave up or concave down? Concave up