1. Unit 4 Lesson 2:Solving Quadratic Equations by Graphing
Advanced Math Topics
Mrs. Mongold

2. Parts of a Quadratic Equation
y = ax2 + bx + c

3. Parts of a Quadratic Equation
y = ax2 + bx + c
ax2 is the quadratic term.

4. Parts of a Quadratic Equation
y = ax2 + bx + c
ax2 is the quadratic term.
bx is the linear term.

5. Parts of a Quadratic Equation
y = ax2 + bx + c
ax2 is the quadratic term.
bx is the linear term.
c is the constant term.

6. Parts of a Quadratic Equation
y = ax2 + bx + c
ax2 is the quadratic term.
bx is the linear term.
c is the constant term.
The highest exponent is two; therefore, the degree is two and there are two solutions

7. Identifying Terms
Example f(x)=5x2-7x+1

8. Identifying Terms
Example f(x)=5x2-7x+1
Quadratic term 5x2

9. Identifying Terms Example f(x)=5x2-7x+1 Quadratic term 5x2
Linear term x

10. Identifying Terms Example f(x)=5x2-7x+1 Quadratic term 5x2
Linear term x
Constant term 1

11. Identifying Terms
Example f(x) = 4x2 - 3

12. Identifying Terms
Example f(x) = 4x2 - 3
Quadratic term 4x2

13. Identifying Terms Example f(x) = 4x2 - 3 Quadratic term 4x2
Linear term

14. Identifying Terms Example f(x) = 4x2 - 3 Quadratic term 4x2
Linear term
Constant term

15. Quadratic Solutions The number of real solutions is at most two.
No solutions
One solution
Two solutions

16. Quadratic Solutions The number of real solutions is at most two.
No solutions
One solution
Two solutions
Number of real solutions is indicated by how many times the graph intercepts x axis

17. 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 referred to as solutions, zeros, or roots.

18. Identifying Solutions
Example f(x) = x2 - 4

19. Identifying Solutions
Example f(x) = x2 – 4
replace y or f(x) with 0 and solve
0 = x2 - 4

20. Identifying Solutions
Example f(x) = x2 – 4
replace y or f(x) with 0 and solve
0 = x2 – 4
Square root property says solutions are x = 2 and x = - 2

21. Identifying Solutions
Example f(x) = x2 – 4
replace y or f(x) with 0 and solve
0 = x2 – 4
Square root property says solutions are x = 2 and x = - 2
Two points aren’t enough we need 5

22. To get 3 more we could make a table or x/y chart….but that’s not fun!

23. So we are going to get our equation in standard form (which is fun) and faster/easier!

24. Standard Form y= a(x – h)2 + k Vertex (h, k) Axis of Symmetry x = h
a tells us how fat/skinny and if parabola is opening up or down

25. Identifying Solutions
Example f(x) = x2 – 4
Standard Form:
y = (x - 0)2 – 4
Vertex (0, -4)
Solutions are -2 and 2 so we graph (-2, 0) and (2, 0)

26. Now we have 3 points we only need 2 more
Use x/y chart one time and the AOS will get us the 5th point
x = 1 gives us y = -3

27. Now graph Vertex (0, -4) Solutions (-2,0) and (2,0) Point (1, -3)
Use AOS x = 0 to get 5th point
on parabola (-1, -3)

28. Identifying Solutions
Now you try this problem.
f(x) = 2x - x2

29. Identifying Solutions
Now you try this problem.
f(x) = 2x - x2
Solutions are 0 and 2.

30. Graphing Quadratic Equations
The graph of a quadratic equation is a parabola.
The roots or zeros are the x-intercepts.
The vertex is the maximum or minimum point.
All parabolas have an axis of symmetry.

31. Graphing Quadratic Equations
One method of graphing uses a table with arbitrary
x-values.
Graph y = x2 - 4x

32. Graphing Quadratic Equations
One method of graphing uses a table with arbitrary
x-values.
Graph y = x2 - 4x
Roots 0 and 4 , Vertex (2, -4) ,
Axis of Symmetry x = 2

33. Graphing Quadratic Equations
Try this problem y = x2 - 2x - 8.
Roots
Vertex
Axis of Symmetry

34. Graphing Quadratic Equations
The graphing calculator is also a helpful tool for graphing quadratic equations.
We will talk about this tomorrow
HW worksheet #1