1. Quadratic Equations and Functions
Algebra I
Algebra I
~ Chapter 10 ~
Quadratic Equations and Functions
Lesson 10-1 Exploring Quadratic Graphs
Lesson 10-2 Quadratic Functions
Lesson 10-3 Finding & Estimating Square Roots
Lesson 10-4 Solving Quadratic Equations
Lesson 10-5 Factoring to Solve Quadratic Equations
Lesson Completing the Square
Lesson Using the Quadratic Formula
Lesson Using the Discriminant
Lesson Choosing a Linear, Quadratic, or Exponential Model
Chapter Review

2. Exploring Quadratic Graphs
Lesson 10-1
Cumulative Review Chap 1-9

3. Exploring Quadratic Graphs
Lesson 10-1
Notes
Quadratic Function – a function in the form ax2 + bx + c, where a ≠ 0.
Examples ~ y = 2x2, y = x2 -7, y = x2 – x – 3
The graph of a quadratic function is a parabola…
The graph of y = x2 is ~> ~> ~>
A parabola can be folded so that the two sides match exactly. The line that divides the parabola into two matched sides is called the axis of symmetry.
The highest or lowest point of a parabola is called its vertex.
If a > 0 in y = ax2 + bx + c ~>
~>
If a < 0 in y = ax2 + bx + c ~>
The vertex is identified as an ordered pair and as minimum or maximum…
The parabola opens upward
The vertex is the minimum point
The parabola opens downward
The vertex is the maximum point

4. Exploring Quadratic Graphs
Lesson 10-1
Notes
Identify the vertex of each graph and tell whether it is a minimum or a maximum.
Graphing y = ax2
Make a table of values.
Graph the points.
(3) Find the corresponding points on the other side of the axis of symmetry.
Graph f(x) = -2x2
The value of a affects the width of the parabola as well as the direction it opens.
You can order quadratic functions by their widths.
Order y = x2, y = ½x2, and y = -2x2 from widest to narrowest…
Graphing y = ax2 + c
The value of c translates the vertex of the graph up (+) or down (-).

5. Exploring Quadratic Graphs
Lesson 10-1
Notes
Graph y = 2x2 + 3
Graph y = -1/2x2 – 4
In summary…
(1) The coefficient of x2, a, determines the width and whether the parabola points upward (+) or downward (-).
The constant, c, determines the vertex location above or below 0.
(3) Ordering quadratic graphs by width, the smaller the coefficient, a, of x2, the wider the graph.

6. Exploring Quadratic Graphs
Lesson 10-1
Exploring Quadratic Graphs
Homework
Homework – Practice 10-1
#1-26

7. Quadratic Functions
Lesson 10-2
Practice 10-1

8. Quadratic Functions
Lesson 10-2
Practice 10-1

9. Quadratic Functions
Lesson 10-2
Practice 10-1

10. Notes Quadratic Functions Lesson 10-2 y = 2x2 + 2x
Graphing y = ax2 + bx + c
y = 2x2 + 2x
(1) Find the axis of symmetry… x = -b/2a Then find the y coordinate. These are the coordinates of the vertex.
(2) Find two other points on the graph.
(3) Reflect those two points over the axis of symmetry. Draw the parabola.
Graph 2x2 + 2x
Graph f(x) = x2 – 6x + 9
Graphing Quadratic Inequalities
y ≤ x2 + 2x – 5
Graph the boundary curve…
Shade the area below the curve because it is less than or equal to.

11. Quadratic Functions
Lesson 10-2
Notes
Graph y > x2 + x + 1

12. Lesson 10-2
Quadratic Functions
Homework
Homework ~ Practice 10-2 even

13. Finding & Estimating Square Roots
Lesson 10-3
Practice 10-2

14. Finding & Estimating Square Roots
Lesson 10-3
Practice 10-2

15. Finding & Estimating Square Roots
Lesson 10-3
Practice 10-2

16. Finding & Estimating Square Roots
Lesson 10-3
Practice 10-2

17. Finding & Estimating Square Roots
Lesson 10-3
Notes
Finding Square Roots
Every positive number has two square roots… The square root of 16 = 4 and -4 or ± 4.
√25 means the positive or principal square root of 25 which is 5. -√25 means the negative square root of 25 which is -5. You can use ± to represent both square roots.
Simplifying Square Root Expressions
√64 = √100 = ±√49 = √1/25 = √121 =
Rational & Irrational Square Roots
Rational square roots have a terminating or repeating decimal…
Irrational square roots have decimals that do not repeat.
±√81 = √8 = √225 = √75 = ±√1/4 =
Estimating Square Roots
You can estimate square roots by using perfect squares. Estimation places the square root between two consecutive integers.

18. Finding & Estimating Square Roots
Lesson 10-3
Notes
Example - √18.5
√16 < √18.5 < √25 so…
4 < √18.5 < 5 so √18.5 is between 4 and 5.
Your turn -√105 is between what two consecutive integers?
Approximating Square Roots with a Calculator
Find √18.5 to the nearest hundredth…
Find √17.81 to the nearest hundredth
Find -√203 to the nearest hundredth

19. Finding & Estimating Square Roots
Lesson 10-3
Finding & Estimating Square Roots
Homework
Homework – Practice 10-3 odd

20. Solving Quadratic Equations
Lesson 10-4
Practice 10-3

21. Solving Quadratic Equations
Lesson 10-4
Notes
Solving Quadratic Equations by Graphing
A quadratic equation is an equation that can be written in the form…
ax2 + bx+ c = 0, where a ≠ 0. This is the standard form of a quadratic equation.
Quadratic equations can have two, one, or no real-number solutions.
Algebra I focusing only on real-number solutions.
Solve by graphing… x2 – (The solution(s) are the x-intercepts)
What about x2 = 0 ?
x2 – 1 = 0
2x2 + 4 = 0
x2 – 16 = -16
Solving Quadratic Equations Using Square Roots
To solve an equation in the form x2 = a; find the square roots of both sides.

22. Solving Quadratic Equations
Lesson 10-4
Notes
t2 – 25 = 0
t2 = 25
t = ± 5
Find the solution(s) 3n = 12
Try… g = 0
Factoring can also be used to solve the quadratic equation…
x2 - 9 = 0
(x + 3) (x – 3) = 0
(x + 3) = 0 or (x – 3) = 0
x = x = Solutions ±3

23. Solving Quadratic Equations
Lesson 10-4
Homework
Homework – Practice 10-4
odd

24. Factoring to Solve Quadratic Equations
Lesson 10-5
Practice 10-4

25. ~ Chapter 10 ~
Algebra I
Algebra I
Chapter Review

26. Chapter 10 Review Part 1
Chapter Review

27. Chapter 10 Review Part 1
Chapter Review

28. Chapter 10 Review Part 1
Chapter Review

29. Factoring to Solve Quadratic Equations
Lesson 10-5
Notes
Using the Zero-Product Property
If ab = 0, then a = 0 or b = 0
Solve (x + 7) (x – 4) = 0
So… x + 7 = 0 or x – 4 = 0
x = or x = 4
Your turn… Solve (3y – 5) (y – 2) = 0
Solving by Factoring
x2 – 8 x – 48 = 0
(x – 12) (x + 4) = 0
x – 12 = 0 or x + 4 = 0
x = or x = -4
Your turn… x2 + x – 12 = 0
x = -4 or x = 3

30. Factoring to Solve Quadratic Equations
Lesson 10-5
Notes
2x2 – 5x = 88
2x2 – 5x – 88 = 0
Your turn… Solve x2 – 12x = -36
x2 – 12x + 36 = 0
(x - 6)2 = 0
x = 6

31. Factoring to Solve Quadratic Equations
Lesson 10-5
Homework
~ Homework ~
Practice 10-5 even

32. Completing the Square
Lesson 10-7
Practice 10-5

33. Using the Quadratic Formula
Lesson 10-7
Notes
Using the Quadratic Formula
If ax2 + bx + c = 0, and a ≠ 0, then…
x = -b ± b2 – 4ac 2a
Make sure your quadratic equation is in standard form…
Solve x2 + 6 = 5x
x2 – 5x + 6 = 0
x = - (-5) ± (-5)2 – 4(1)(6)
2(1)
x = 5 ± – 24 2
x = 5 ± √ 1 = or 5 – 1 = or = or 2

34. Using the Quadratic Formula
Lesson 10-7
Notes
Your turn…
Solve using the quadratic formula
x2 – 4x = 117
x2 – 2x – 8 = 0
Finding Approximate Solutions
2x2 + 4x – 7 = 0
x = - (4) ± (4)2 – 4(2)(-7) = -4 ± – (-56)
2(2) 4
x = -4 + √72 or √72 ≈ or ≈ or -3.12
7x2 – 2x – 8 = 0

35. Using the Quadratic Formula
Lesson 10-7
Homework
Homework – Practice 10-7 even

36. Using the Discriminant
Lesson 10-8
Practice 10-7

37. Using the Discriminant
Lesson 10-8
Notes
Number of Real Solutions of a Quadratic Equation
Discriminant – The expression under the radical in the quadratic formula (b2 – 4ac) The discriminant can be used to determine how many solutions a quadratic equation has before you solve it…
If b2 – 4ac > 0, there are 2 solutions
If b2 – 4ac = 0, there is 1 solution
If b2 – 4ac < 0, there are no solutions
Using the Discriminant
Find the number of solutions for x2 – 2x – 3
b2 – 4ac = (-2)2 – 4(1)(-3)
4 – (-12) = = 16 > 0 , so there are 2 solutions.
Your turn… Find the number of solutions for 3x2 – 4x – 7
Find the number of solutions for 5x2 + 8 = 2x

38. Using the Discriminant
Lesson 10-8
Homework
Homework – Practice 10-8 odd
&
Chapter 10 Review Part 2

39. Using the Discriminant
Lesson 10-8
Homework

40. ~ Chapter 10 ~
Algebra I
Algebra I
Chapter Review

41. ~ Chapter 10 ~
Algebra I
Algebra I
Chapter Review