1. Quadratic Functions and Their Graphs
Lesson 1-7
Quadratic Functions and Their Graphs

2. Objective:

3. To define and graph quadratic functions.
Objective:
To define and graph quadratic functions.

4. Quadratic Function:

5. Quadratic Function:
f(x) = ax2 + bx +c where a 0.

6. The graph of a quadratic function is called a parabola.
f(x) = ax2 + bx +c where a 0.
The graph of a quadratic function is called a parabola.

7. Special Parts of a Parabola:

8. Special Parts of a Parabola:
Vertex:
The turning point. It is either a maximum or minimum.

9. Special Parts of a Parabola:
Axis of Symmetry:
A vertical line that passes through the vertex.

10. Special Parts of a Parabola:
Axis of Symmetry:
This line is midway between the x-intercepts therefore it is the “average” of the x-values.

11. Axis of Symmetry:

12. Can always be found by calculating the formula
Axis of Symmetry:
Can always be found by calculating the formula

13. Vertex:

14. Can always be found using the formula
Vertex:
Can always be found using the formula

15. Discriminant:

16. Discriminant: If b2 – 4ac > 0
Parabola crosses x-axis twice. There will be two x-intercepts.

17. Discriminant: If b2 – 4ac > 0
Parabola crosses x-axis twice. There will be two x-intercepts.
If b2 – 4ac = 0
Parabola is tangent to the x-axis. There is only one x-intercept.

18. Discriminant: If b2 – 4ac > 0
Parabola crosses x-axis twice. There will be two x-intercepts.
If b2 – 4ac = 0
Parabola is tangent to the x-axis. There is only one x-intercept.
If b2 – 4ac < 0
Parabola never crosses the x-axis so there are no x-intercepts.

19. Find the intercepts, axis of symmetry, and the vertex of the given parabola. y = (x + 4)(2x – 3)

20. Now sketch the graph.

21. Sketch the graph of the parabola
Sketch the graph of the parabola. Label the intercepts, the axis of symmetry, and the vertex. y = 2x2 – 8x + 5

22. If the equation can be written in the form of :

23. If the equation can be written in the form of :
then the vertex of the parabola is (h, k) and the axis of symmetry is the equation x = h.

24. Find the vertex of the parabola by completing the square
Find the vertex of the parabola by completing the square. y = -2x2 + 12x + 4

25. Now, find the x- and y-intercepts. y = -2x2 + 12x + 4

26. Find the equation of the quadratic function f with f(-1) = -7 and a maximum value f(2) = -1. Show that the function has no x-intercepts.

27. Where does the line y = 2x + 5 intersect the parabola y = 8 – x2
Where does the line y = 2x + 5 intersect the parabola y = 8 – x2? *Show this both algebraically and graphically.

28. Find an equation of the function whose graph is a parabola with x-intercepts 1 and 4 and a y-intercept of -8.

29. General Linear Function:
So far we know
f(x) = dx + e
but we also know
y = mx + b
where m = slope.

30. What is the slope of y = ¼ x - 3?

31. Assignment: Pgs. 40-41 C.E. 1-6 all W.E. 1-25 (Left Column)