1. Entry Task

2. 4.1 Graphing Quadratic Functions
Learning Target: I can Identify and graph quadratic equations.

3. Graphs of Quadratic Functions
Axis of symmetry
Important features of graphs of parabolas
Maximum
x-intercepts
Minimum
Vertex

4. Graphing Quadratics
If you were asked to graph a quadratic, what information would you need to know to complete the problem?
The vertex, because we need to know where the graph is located in the plane
If the parabola points up or down, and whether it opens normal, narrow or wide
Our graphs will be more “quick sketches” than exact graphs.

5. Graph of f(x)=x2 Axis of symmetry is x = 0 x f(x) 1 -1 2 -2 41 -1 2 -2 4
Points up, opens “normal”
Notice the symmetry
Vertex at (0, 0)
The “mother function”

6. More with Vertex Form
The vertex is (h, k). Changes in (h, k) will shift the quadratic around in the plane (left/right, up/down).
The axis of symmetry is x = h
If a > 0, the graph points up
If a < 0, the graph points down
Example #2
Example #1
Vertex is (4, 0)
Axis is x = 4
Points down
Vertex is _____
Axis is _______
Points _______
Vertex is (0, 6)
Axis is x = 0
Points up
Vertex is _____
Axis is _______
Points _______
Example #3
Vertex is (-3, -1)
Axis is x = -3
Points up
Vertex is _____
Axis is _______
Points _______
Notice that you take the opposite of h from how it is written in the equation

7. Equations of Quadratic Functions
Vertex Form
Standard Form

8. More about a When a = 1, the graph is “normal” a =1 a =1/5 a = 5
What happens to the graph as the value of a changes?
If a is close to 0, the graph opens _______________
If a is farther from 0, the graph opens ____________
If a > 0, the graph points________
If a < 0, the graph points ________
If a is close to 0, the graph opens wider
If a is farther from 0, the graph opens narrower
If a > 0, the graph points up
If a < 0, the graph points down

9. Max and Min Problems
What is the definition of the maximum or minimum point of a quadratic function?
The vertex of a quadratic function is either a maximum point or a minimum point
max
min
If a quadratic points down, the vertex is a maximum point
If a quadratic points up, the vertex is a minimum point
If you are asked to find a maximum or minimum value of a quadratic function, all you need to do is find its vertex

10. Sketch each quadratic
Identify the vertex, axis of symmetry, the max or min value, and domain and range.
AOS: x = -3 Min: k =-1 D: all reals R: all reals > - 1
V = (-3, -1)
Points up
Narrow
AOS: x = 0
Max: k = 4
D: all reals
R: all reals < 4
V = (0, 4)
Points down
Wide

11. Sketch each quadratic V = (-3, -1) V = (2, 1) Points up Points down
Narrow
V = (2, 1)
Points down
Normal
V = (-4, 2)
Points up
Normal
V = (0, 4)
Points down
Wide

12. Homework
Homework – p. 199 #9-36 by 3’s
Challenge - 56

13. Summary: Be able to compare and contrast vertex and standard form
Vertex Form
Standard Form
How do you find the Vertex?
How do you find the Axis of Symmetry?
How can you tell if the function:
points up or down?
opens normal, wide or narrow?
What info is needed to do a quick sketch or graph?
How do you find the solutions? (x-intercepts, roots, zeroes, value of x when y = 0)
Set = 0, get “squared stuff” alone, then use square root method
Set = 0 and use method of choice (factor, formula or square root)

14. More with Standard Form
To find the x-value of the vertex, use the formula
To find the y-value, plug in x and solve for y
The axis of symmetry is
If a > 0, the graph points up
If a < 0, the graph points down
Example #1
b = 4, a = -1
Find x-value of vertex using formula
Vertex is _____
Axis is _______
Points _______
Vertex is (2, 1)
Axis is x = 2
Points down
Find y-value using substitution 2
(2)
(2)

15. More examples Example #2 You try: Vertex is _____ Axis is _______
Points _______
Find x-value using formula
b = -1, a = 3
Find y-value using substitution 16 16 16
Vertex is (3, 17)
Axis is x = 3
Points down
Vertex is _____
Axis is _______
Points _______
Vertex is (1/6, 59/12)
Axis is x = 1/6
Points up

16. Finding x-intercepts of quadratic functions
What are other words for x-intercepts?
Name 4 methods of finding the x-intercepts of quadratic equations:
roots
zeroes
solutions
All are the value of x when y = 0
factoring
The Square Root
The Quadratic Formula
Graphing

17. Example
An object is thrown upward from the top of a 100 foot cliff. Its height in feet about the ground after t seconds given by the function f(t) = -16t2 + 8t
What was the maximum height of the object?
How many seconds did it take for the object to reach its max height?
How can we find the answer? What is the question asking for?

18. vertex Example What was the maximum height of the object?
How many seconds did it take for the object to reach its maximum height?
What is the definition of the maximum or minimum point of a quadratic function?
The vertex of a quadratic function is either a maximum point or a minimum point
vertex

19. Example f(t) = -16t2 + 8t + 100. f(1/4) = -16(1/4)2 + 8(1/4) + 100.
Step 2: Understand the equation
Example y x
Input: time
Output: height
Step 1: Visualize the problem
f(t) = -16t2 + 8t
To find the max values, find the vertex
The x-value of the vertex is the max time
(1/4, 101)
It took about .25 seconds for the object to reach its max height
The y-value of the vertex is the max height
f(1/4) = -16(1/4)2 + 8(1/4)
f(t) = -16t2 + 8t
The max height was 101 feet
f(1/4) = 101