1. Quadratic Functions (Graphing)
Algebra 2A – Unit 2
Quadratic Functions
(Graphing)

2. 2-1: Graphing Quadratic Functions Learning Targets:
I can graph quadratics using five points.
I can find and interpret maximum and minimum values.

3. Picture/Formula In your own words:
Term
Picture/Formula
In your own words:
Function
Quadratic Function
Standard Form:
Parabola
Vertex
Max/Min
x-coordinate
of vertex
Axis of symmetry
y-intercept
Graph of a quadratic function:
Plug x into the
function to find y.
This is where the
graph makes its turn.

4. Using the TI to Graph Know these function keys on your calculator: y =
set window
table
max and min
Choose some values around the axis of symmetry, so around the x=2 point

5. Example 1: Graph f(x) = 3x2 - 6x + 7
Axis of symmetry:
y-intercept:
Direction of opening: Up / down?
“a” value?
Maximum / Minimum? Value: _____________
Minimum of x=1
Now plot the points: x
f(x) -1 1 2 3
Vertex:
(1, 4) 16 7 4 7 16

6. Example 2: Graph f(x) = - x 2 + 6x - 4
Now plot the points:
Axis of symmetry:
y-intercept:
Direction of opening: Up / down?
“a” value?
Maximum / Minimum? Value: _____________
Maximum of x=3 x
f(x) 1 2 3 4 5
Vertex:
(3, 5) 1 4 5 4 1

7. Lesson 2.1: Closing H.W. Practice 2.1
Can you graph quadratics using five points and/or your calculator?
Can you find max. and min. values?
H.W. Practice 2.1

8. Click the mouse button or press the Space Bar to display the answers.
Transparency 2

9. Transparency 2a

11. Analyzing graphs of Quadratic Functions
Algebra 2A – Unit 2 Lesson 2.2
Analyzing graphs of Quadratic Functions

12. Lesson 2.2 Learning Targets:
I can graph quadratic functions in vertex form.
I can write quadratic functions in standard and/or vertex form.

13. Picture/Formula In your own words: Vertex Form Standard Form
Term
Picture/Formula
In your own words:
Quadratic Function
Vertex Form:
Vertex
Max/Min
x-coordinate
of vertex
Axis of symmetry
Vertex Form
Standard Form

14. Exploration Activity

15. Example 1: Graph f(x) = (x – 3)2 – 2
Now plot the points:
Direction of opening: Up / down ?
Vertex: ________
Max / Min ?
Axis of symmetry: ___________ x
f(x)

16. Example 2: Graph f(x) = ½ (x + 5)2 + 3
Direction of opening: Up / down ?
Vertex: ________
Max / Min ?
Axis of symmetry: ___________ x
f(x)

17. Your Turn 1: Graph f(x) = 4(x + 3)2 – 2
Direction of opening: Up / down ?
Vertex: ________
Max / Min ?
Axis of symmetry: ___________ x
f(x)

18. Your Turn 2: Graph f(x) = -2 (x + 1)2 + 4
Direction of opening: Up / down ?
Vertex: ________
Max / Min ?
Axis of symmetry: ___________ x
f(x)

19. Example 3: Plug in values for h, k and x, y to find a.
Plug in values for h, k and x, y to find a.
Rewriting the equation in standard form.
Now look at this:
-20 = a(0 – 4)2 - 36
Next, solve for a.
-20 = 16a - 36
y = 1(x – 4)2 - 36
16a = 16
y = 1(x– 4)(x – 4) - 36
a = 1
y = 1(x2 -8x +16)- 36
Finally, rewrite the equation using a, h and k.
y = x2 -8x - 20
y = 1(x – 4)2 - 36
Same parabola in different forms.

20. Example 4: Plug in values for h, k and x, y to find a. 2 = a(0+3)2 - 1
Plug in values for h, k and x, y to find a.
2 = a(0+3)2 - 1
Next, solve for a.
2 = 9a - 1
3 = 9a
Finally, rewrite the equation using a, h and k.

21. Your Turn 3: Plug in values for h, k and x, y to find a.
Plug in values for h, k and x, y to find a.
0 = a(2 - 3)2 +- 1
Next, solve for a.
0= a - 1
1 = a
Finally, rewrite the equation using a, h and k.
y = 1(x - 3)2 - 1

22. How high does the object go? ___________________
Use your Graphing Calculator to solve the following problems:
Word Problem 1: An object is propelled upward from the top of a 500 foot building. The path that the object takes as it falls to the ground can be modeled by h = -16t2 +100t where t is the time (in seconds) and h is the corresponding height of the object. The velocity of the object is v = -32t +100 where t is seconds and v is velocity of the object
y =
How high does the object go? ___________________
When is the object 550 ft high? __________________
With what velocity does the object hit the ground? __________________

23. How many seconds is the rock in the air?
Use your Graphing Calculator to solve the following problems:
Word problem 2: An astronaut standing on the surface of the moon throws a rock into the air with an initial velocity of 27 feet per second. The astronaut’s hand is 6 feet above the surface of the moon. The height of the rock is given by h = -2.7t2+ 27t + 6
y =
How many seconds is the rock in the air?
How high did the rock go?

24. Lesson 2.2: Assignment Check for Understanding: Closure Assignment:
Practice 2.2

25. Click the mouse button or press the Space Bar to display the answers.
Transparency 7

26. Transparency 7a

28. Solving Quadratic Equations by graphing
Algebra 2A – Unit 2 Lesson 2.3
Solving Quadratic Equations by graphing

29. Lesson 2.3 Learning Targets:
I can solve quadratic equations by graphing. (exact roots)
I can estimate solutions of quadratic equations by graphing. (approximate roots)

30. Vocabulary Term Picture/Formula In your own words: Quadratic Equation
Zeros
Roots
x- intercepts of the graph
solutions of the quadratic equation
Cases:
two real roots
one real root
no real roots , Ø

31. Example: Not a point but the 2 places where
Not a point but the 2 places where
the graph crosses the x-axis.
This is called a double root.
-3 is the answer twice.

32. Part 1 : Exact roots Example 1:
Old School or Vintage style:
Solve x2 + x – 6 = 0 by graphing.
Vertex: ________ -3 x
f(x) -2 -1 1 2 -4 -6 -6 -4
Exact Roots of the equation (or zeros of the function): ________ & _______ -3 2

33. Part 1 : Exact roots Your Turn 1:
Old School or Vintage style:
Solve x2 - 4x – 5 = 0 by graphing.
Vertex: ________ -1 x
f(x) 1 2 3 4 5 -5 -8 -9 -8 -5
Exact Roots of the equation (or zeros of the function): ________ & _______ -1 5

34. Part 2 : Approximate roots Example 2:
Use your Graphing Calculator to solve the following problems:
Part 2 : Approximate roots Example 2:
Solve x2 - 2x – 2 = 0 by graphing.
Look under the table in the calculator.
Vertex: ________ x
f(x) 2 -2 -1 3 1 -3
Approximate roots: _____________________________

35. Part 2 : Approximate roots Your Turn 2:
Use your Graphing Calculator to solve the following problems:
Part 2 : Approximate roots Your Turn 2:
Solve x2 - 4x + 4 = 0 by graphing.
Vertex: ________ x
f(x) 1 3 4 2
Double Root of x=2
Approximate roots: _____________________________

36. Their sum is 4, and their product is -12.
Use your Graphing Calculator to graph and find the roots:

37. Lesson 2.3: Assignment Check for Understanding: Closure Assignment:
Practice 2.3

38. Click the mouse button or press the Space Bar to display the answers.
Transparency 3

39. Transparency 3a