4. 1.Two forms of a quadratic equation 2.Review: Graphing using transformations 3.Properties of the graph 4.Graphing by hand a)Method 1: Use standard form Complete the Square a)Method 2: Use quadratic form: 5.Identify the vertex and axis of symmetry 6.Find maximum or minimum of applied quadratic problems Section 3-1 Quadratic Functions

5. 1. Two forms of a quadratic Functions 1. A quadratic function is a function of the form: where a, b, and c are real numbers and a ≠ 0. 2. Standard (Vertex) Form of a quadratic function:

6. 2. Review: Graphing using Transformations 1.Identify the x and y intercepts 2.Identify the parent function 3.Describe the sequence of transformations 4.graph 5.Does this function have a maximum or minimum? What is it?

7. Opens up Opens down 3. Properties of the graph Axis of Symmetry Vertex Axis of Symmetry Maximum at vertex Minimum at vertex

8. Method 1: Graph standard (vertex) form of quadratic: Direction of parabola: If a> 0 then graph opens up; If a<0 then graph opens down Vertex : x-intercepts: Solve: y-intercept: Axis of Symmetry: ( points on the graph are equidistant horizontally from x=h ) Practice:

9. 4 a)Complete the Square Complete the square of to rewrite in standard form: Determine the vertex and axis of symmetry for the quadratic:

10. 4 b) Method 2: Find vertex directly Axis of Symmetry: Vertex: Find y-intercepts Find the x-intercepts. Additional point using symmetry. a > 0 : opens up a < 0 : opens down

11. Let’s try the previous example again, using Method 2. 5. The Discriminant What does the discriminant tell you? # x-intercepts

12. Example: Two Ways to Graph Quadratic Functions by hand Which method do you prefer for graphing:

13. 6. a) Determine the quadratic equation: Given the vertex and a point Determine the quadratic equation whose vertex is (4,-1) and passes through the point (2,7). Recall standard form:

14. 6 b) Determine the quadratic equation: Given x-intercepts and a point Suppose and r1 and r2 are x-intercepts. We can write Example: Suppose a quadratic has zeros at -3 and 5. And suppose the function passes through the point (6,63). Write the quadratic.

15. 7. Properties of the graph. Max/Min is at: and Domain: Range: If a > 0, then Range= If a < 0, then Range= Increasing/Decreasing:

16. Application Example: A farmer has 2000 ft of fencing to enclose a rectangular area that borders a river. The fencing will be along the 3 sides other than the river. a)Express area as a function of one variable. b) Determine the dimensions that will maximize the area.

17. Maximums and Minimums An engineer collects the following data showing the speed s of a Ford Taurus and its average miles per gallon M. Speed, sMiles/Gallon, M 3018 3520 4023 4025 4525 5028 5530 6029 6526 6525 7025 a) Determine a scatter plot for the data b) Use a graphing utility to find the quadratic function of best fit to this data c) Use the function found to determine the speed that maximizes miles per gallon d) Use the function to predict miles per gallon for a speed of 63 mph.

18. Section 3.1 p. 164 # 29,35,42,49,53,55,59,61,67,(69),71,78,79,81