1. Quadratic Functions

2. What is a Quadratic Function?
An equation that can be written in the form
ax2 + bx + c = 0
It is a 2nd degree polynomial
Does not contain an exponent higher than a 2.

3. Quadratic Function f(x) = ax2 + bx + c
Identify the a, b, and c values in each example.
Example 1: 2x2 + 3x + 10
Example 2: -3x2 + 5x + 21
Example 3: -½x2 – 8x + 7
Example 4: 10x2 – 9x – 3
Example 5: -x2 + ¼x – ½
Example 6: x2 - 11x + 12

4. Let’s Review what we already know 
Let’s Review what we have learned about the A,B, C’s of Quadratics:
What does the A value do to the graph?
What does the B value do to the graph?
What does the C value do to the graph?
4/5/2019
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5. Graphs of Quadratic Functions
Let’s take a look at several graphs of quadratic functions and investigate how changing the coefficients of the variables in the equation effects the graph. Reminder: Quadratic Equation y = ax2 + bx + c
4/5/2019
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6. Quadratic Functions The graph of a quadratic function is a parabola.y x
The graph of a quadratic function is a parabola.
Vertex
A parabola can open up or down.
If the parabola opens up, the lowest point is called the vertex.
If the parabola opens down, the vertex is the highest point.
Vertex
NOTE: if the parabola opened left or right it would not be a function!

7. Standard Form a > 0 a < 0y x
The standard form of a quadratic function is
a < 0
a > 0
y = ax2 + bx + c
The parabola will open up when the a value is positive.
The parabola will open down when the a value is negative.

8. The axis of symmetry ALWAYS passes through the vertex.
Parabolas have a symmetric property to them.
If we drew a line down the middle of the parabola, we could fold the parabola in half.
We call this line the axis of symmetry.
Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other side.
The axis of symmetry ALWAYS passes through the vertex.

9. What are we doing today??? Are you Math Ready???? Let’s get started!
We have already analyzed quadratic graphs and visually determined the vertex point.
Now we are going to:
Calculate the vertex algebraically.
Define and represent the axis of symmetry.
Define and represent the x-intercepts
Define and represent the y-intercept
Are you Math Ready????
Let’s get started!

10. Consider the following quadratic function: f(x) = x2 + 2x – 3
Let’s talk about several important parts of a quadratic function: Where is the vertex?
(-1, -4)

11. Consider the following quadratic function: f(x) = x2 + 2x – 3
Let’s talk about another important part of a quadratic function: How do we algebraically calculate the vertex?

12. Consider the following quadratic function: f(x) = x2 + 2x – 3
Calculating the vertex.
The vertex is a coordinate point (x, y) on the graph, now that we have the x value how do you think we determine the y value?

13. Consider the following quadratic function: f(x) = x2 + 2x – 3
Calculating the vertex.
Substitute the value of x into the given function equation above and solve! The answer is the value for y.
When x = -1, y = -4. Vertex is: (-1, -4).

14. Consider the following quadratic function: f(x) = x2 + 2x – 3
Let’s talk about another important part of a quadratic function: What is the axis of symmetry?
Now that you see what it is, how would you define the axis of symmetry?

15. Consider the following quadratic function: f(x) = x2 + 2x – 3
Let’s talk about another important part of a quadratic function: How do we represent this axis of symmetry?
x = -1

16. Consider the following quadratic function: f(x) = x2 + 2x – 3
Let’s talk about another important part of a quadratic function:
Where are the x-intercepts?
Where does the function cross the x-axis?
x-intercepts:
(1, 0) & (-3, 0)

17. Consider the following quadratic function: f(x) = x2 + 2x – 3
Let’s talk about another important part of a quadratic function:
Where is the
y-intercept?
Where does the function cross the y-axis?
y-intercept:
(0, -3)

18. Quadratic Functions and their important parts!
What important parts do you recognize in this graph?

19. Quadratic Functions and their important parts!
What important parts do you recognize in this graph?

20. Quadratic Functions and their important parts!
What important parts do you recognize in this graph?

21. Quadratic Functions and their important parts!
What important parts do you recognize in this graph?

22. Quadratic Functions and their important parts!
What important parts do you recognize in this graph?
y = x2 – 3x – 10

23. Let’s Do It Again Ourselves
Let’s Do It Again Ourselves!! Consider the following quadratic function: f(x) = x2 – 2x – 15
Where is the vertex? Algebraically calculate the vertex.
(1, -16)

24. Consider the following quadratic function: f(x) = x2 – 2x – 15
Where is the axis of symmetry? Draw in the axis of symmetry. What is the axis of symmetry?

25. Consider the following quadratic function: f(x) = x2 – 2x – 15
Where are the x-intercepts?
Where does the function cross the x-axis?
x-intercepts:
(-3, 0) & (5, 0)

26. Consider the following quadratic function: f(x) = x2 – 2x – 15
Where is the y-intercept?
Where does the function cross the y-axis?
y-intercept:
(0, -15)

27. Let’s Do It Again Ourselves
Let’s Do It Again Ourselves!! Consider the following quadratic function: f(x) = x2 + 3x
Where is the vertex? Algebraically calculate the vertex.
(-1.5, -2.25)

28. Consider the following quadratic function: f(x) = x2 + 3x
Where is the axis of symmetry? Draw in the axis of symmetry. What is the axis of symmetry?

29. Consider the following quadratic function: f(x) = x2 + 3x
Where are the x-intercepts?
Where does the function cross the x-axis?
x-intercepts:
(-3, 0) & (0, 0)

30. Consider the following quadratic function: f(x) = x2 + 3x
Where is the y-intercept?
Where does the function cross the y-axis?
y-intercept:
(0, 0)

31. Now, Visualize the graph!
Given: f(x) = x2 – 4x + 3
Open up or down?
Calculate the vertex?
What is the axis of symmetry?
Where is the y-intercept?

32. Now, Visualize the graph!
Given: f(x) = 2x2 + 3x – 1
Open up or down?
Calculate the vertex?
What is the axis of symmetry?
Where is the y-intercept?

33. Now, Visualize the graph!
Given: f(x) = 5x2 – 2x + 5
Open up or down?
Calculate the vertex?
What is the axis of symmetry?
Where is the y-intercept?

34. Now, Visualize the graph!
Given: f(x) = x2 – 2x – 15
Open up or down?
Calculate the vertex?
What is the axis of symmetry?
Where is the y-intercept?

35. Ticket Out The Door Homework
Complete the ticket out the door problem. Please hand it to me as you walk out of the door.
Homework
Complete the worksheet for homework.