1. Quadratic Equations and Functions
Chapter 9
Quadratic Equations and Functions

2. 9.1 Graph y = ax2 + c (b = 0)

3. Quadratic Functions parabola The graph of a quadratic function is a:y x
A parabola can open up or down.
Vertex
If the parabola opens up, the lowest point is called the vertex (minimum).
Let students know that in Algebra I we concentrate only on parabolas that are functions; In Algebra II, they will study parabolas that open left or right.
If the parabola opens down, the vertex is the highest point (maximum).
Vertex
NOTE: if the parabola opens left or right it is not a function!

4. y = ax2 + bx + c The standard form of a quadratic function is:
The parabola will open up when the a value is positive.
a < 0
a > 0
The parabola will open down when the a value is negative.
Remind students that if ‘a’ = 0 you would not have a quadratic function.

5. The Axis of symmetry ALWAYS passes through the vertex.
Parabolas are symmetric.
If we drew a line down the middle of the parabola, we could fold the parabola in half. y x
Axis of Symmetry
We call this line the Axis of symmetry.
The Axis of symmetry ALWAYS passes through the vertex.

6. Steps to Graphing Quadratic Functions
1) Find the Axis of symmetry using:
2) Find the vertex by using x to find y
3) Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve.

7. Graph y = x2 (parent function)

8. Graph y = -3x2 y x

9. Graph y = ½x2 + 1 y x

10. Homework:
9.1 Practice

11. 9.2 Graph y = ax2 + bx + c

12. Find the Axis of symmetry for y = 3x2 – 18x + 7
y = ax2 + bx + c,
When a quadratic function is in standard form
the equation of the Axis of symmetry is
Find the Axis of symmetry for y = 3x2 – 18x + 7
Discuss with the students that the line of symmetry of a quadratic function (parabola that opens up or down) is always a vertical line, therefore has the equation x =#. Ask “Does this parabola open up or down?
a = 3 b = -18
The Axis of symmetry is x = 3.

13. Vertex X-coordinate Finding the Vertex
The Axis of symmetry always goes through the _______. Thus, the Axis of symmetry gives us the ____________ of the vertex.
Vertex
X-coordinate
Find the vertex of y = -2x2 + 8x - 3
STEP 1: Find the Axis of symmetry
STEP 2: Substitute the x – value into the original equation to find the y –coordinate of the vertex.

14. Graphing a Quadratic Function
There are 3 steps to graphing a parabola in standard form.
STEP 1: Find the Axis of symmetry using:
STEP 2: Find the vertex
STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve.
MAKE A TABLE
using x – values close to the Axis of symmetry.

15. Graphing a Quadratic Functiony x
STEP 1: Find the Axis of symmetry
STEP 2: Find the vertex
STEP 3: Make table of values around vertex

16. Example: Graph y = -x2 – 2x + 1

17. Homework:
p.580 #1, 2, 3 – 35odd

18. 9.3 Solve Quadratic Equations by Graphing

19. Solving Equations
When we talk about solving these equations, we want to find the value of x when y = 0. These values, where the graph crosses the x-axis, are called the x- intercepts.
These values are also known as solutions, zeros, or roots.

20. Solving a Quadratic The number of real solutions is at most two.
The x-intercepts (when y = 0) of a quadratic function are the solutions to the related quadratic equation.
The number of real solutions is at most two.
Remind students that x-intercepts are found by setting y = 0 therefore the related equation would be ax2+bx+c=0. Also state that since the highest degree of a quadratic is 2, then there are at most 2 solutions. For the first graph ask “why are there no solutions?”-- there are no solutions because the parabola does not intercept the x-axis. 2nd and 3rd graph ask students to state the solutions. Additional Vocab may be itroduced: The x-intercepts are solutions, zero’s or roots of the equation.
Two solutions
X= -2 or X = 2
One solution
X = 3
No solutions

21. Identifying Solutions
Find the solutions of 2x - x2 = 0
The solutions of this quadratic equation can be found by looking at the graph of f(x) = 2x – x2
The x-intercepts(or Zeros) of f(x)= 2x – x2
are the solutions to 2x - x2 = 0
Point out to students that the function can also be written as y = -x2+2x.
X = 0 or X = 2

22. Example: Solve the equation x2 + 5x + 6 = 0 by graphing

23. Homework:
p. 589 #1, 2, 3 – 39 odd

24. 9.4 Use Square Roots to Solve Quadratic Equations
I can solve a quadratic equation by finding square roots
I can solve a problem about a falling object
CC9-12.A.REI.4b

25. When b = 0, the equation becomes y = ax2 + c
You can use Square Roots Method to solve the equation (extracting roots)

26. Steps to Solving Quadratics Using Square Roots Method
Make sure b = 0
Get variable term on one side, constant on the other
Get variable by itself
Take square root of both sides

27. Example: Solve 2x2 = 8

28. Example: x = 25
Example: x = 7

29. Example: A pinecone drops from a tree 150 feet up
Example: A pinecone drops from a tree 150 feet up. How long will it take the pinecone to reach the ground.

30. Homework:
p. p.597 #1, 2, 3 – 49 odd, 56,

31. 9.6 Solve Quadratic Equations by Using Quadratic Formula
I can solve quadratic equations using the quadratic formula.
CC.9-12.A.REI.4b

32. Quadratic Formula:

33. Example: Solve 3x2 + 5x = 8

34. Example: Solve 2x2 – 7 = x

35. Example: x2 – 8x + 16 = 0

36. Methods for Solving Quadratic Equations
Factoring – good when quadratic eqn can be factored easily
Graphing – use when approximate solutions are fine
Square roots method – use when b = 0 (no bx term)
Quadratic formula – can be used at any time!!!
Some methods are better than others depending on situation

37. Example: tell what method you would use to solve the quadratic equation. Explain.
x2 + 5x + 6 = 0

38. Homework:
p.616 #1, 2, 3 – 43 odd, 47, 48

39. 9.6 extended – The Discriminant

40. 9.5 Solve Quadratic Equations by Completing the Square

41. 9.7 Solve Systems of Quadratic Equations

42. 9.8 Compare Linear, Exponential, and Quadratic Functions

43. 9.9 Model Relationships