1. 7.1. Graphing Quadratics Standard Form
College Algebra
7.1. Graphing Quadratics Standard Form

2. Do Now: Solve each of the following problems 8+3 𝑥+2 2 =44 − 𝑥 2 =4𝑥
8+3 𝑥+2 2 =44
− 𝑥 2 =4𝑥
5 𝑥 2 =35
2 𝑥 2 +7𝑥+1=5
4 𝑥 2 −2𝑥−5=0

3. Do Now:
Complete do now handed to you when you entered class

4. Homework Questions? Comments? Confusions? Concerns?
ASK ASK ASK ASK ASK!

5. Goal:
We currently know how to factor quadratics or things that look like 𝑎 𝑥 2 +𝑏𝑥+𝑐, but… what do they look like on a graph?

6. We know… Linear Equations
𝑦=𝑚𝑥+𝑏

7. We know… Exponential Equations

8. Now… Quadratic Equations
The graph of 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 will always look like a U shape

9. Parabola:
This U shape, is called a parabola

10. Parts of a Parabola

11. Parts of a Parabola
Vertex: The highest or lowest point on a graph

12. Parts of a Parabola
Axis of Symmetry: The vertical “mirror” of the graph that goes through the vertex. Everything on the left side of this line is mirrored on the right side.

13. Parts of a Parabola
Y Intercept: Where the graph crosses the 𝑦−𝑎𝑥𝑖𝑠. There is always one and only one Y intercept for a quadratic equation.

14. Parts of a Parabola
X Intercept: Where the graph crosses the 𝑥- axis. These are the solutions of the quadratic equation:
If the equation has two solutions, it intersects the 𝑥−𝑎𝑥𝑖𝑠 twice.
If the equation has one solution, it intersects the 𝑥−𝑎𝑥𝑖𝑠 once.
If the equation has no solutions, it never intersects the 𝑥−𝑎𝑥𝑖𝑠

15. Visually: X intercepts
Two Intercepts:

16. Visually: One Intercept
One X Intercept:

17. Visually: X Intercepts
No X Intercepts:

18. Realize:
Parabolas can be open “up” (Happy Face) or “down” (Frowny Face)

19. Example One: Graph 𝑓 𝑥 = 𝑥 2 −4𝑥+3 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

20. How to Graph Parabolas:
** Our equation must be in the form 𝑎 𝑥 2 +𝑏𝑥+𝑐=0
Step 1:
We determine the direction of opening by checking if 𝑎 is positive or negative.

21. How to Graph Parabolas:
Step Two:
We find the vertex by…
Find the value of − 𝑏 2𝑎 . This is the x coordinate of the vertex
Plugging in this value for x in the equation yields the y coordinate of the vertex
The axis of symmetry is 𝑥=− 𝑏 2𝑎
We plot the point we found and the axis of symmetry line

22. How to Graph Parabolas:
Step Three:
We find the y-intercept by plugging in zero for x. Plot our value on the y axis.
If the point does not fit in your picture, do not graph it
If the point does fit, graph the point and its mirror image on the other side of your axis of symmetry.

23. How to Graph Parabolas:
Step Four:
Find the x intercepts by solving the quadratic equation. Make 𝑦=0 and you can…
Take the square root if 𝑏=0
Factor the GCF if 𝑐=0
Use PS2 if all three terms are present
Quadratic Formula is all else fails

24. How to Graph Parabolas:
Step Five:
Draw your parabola. You must have at least three points plotted.
Make sure that it is a U shape and not a V shape or straight line.

25. Example One: Graph 𝑓 𝑥 = 𝑥 2 −4𝑥+3 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

26. Example Two: Graph 𝑓 𝑥 = −𝑥 2 −8𝑥−12 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

27. Example Three: Graph 𝑓 𝑥 = 𝑥 2 −5𝑥 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

28. Do Now:
Handed to you when you entered class

29. Example Four: Graph 𝑓 𝑥 = 𝑥 2 −4 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

30. You Try #1: Graph 𝑓 𝑥 = 𝑥 2 −8𝑥+15 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

31. You Try #2: Graph 𝑓 𝑥 =− 𝑥 2 +9𝑥−20 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

32. Realize:
So far everything we have graphed has had two x intercepts! BUT there could be parabola’s with ONE x intercept!

33. Example Five: Graph 𝑓 𝑥 =−4 𝑥 2 +12𝑥−9 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

34. Example Six: Graph 𝑓 𝑥 =4 𝑥 2 −4𝑥+1 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

35. Example Seven: Graph 𝑓 𝑥 =− 𝑥 2 +6𝑥−9 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

36. Realize:
To get another point, choose an x value close to the vertex and plug it in to get the y value! (x,y) gets you a point!

37. Example Eight: Graph 𝑓 𝑥 =9 𝑥 2 +36𝑥+36 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

38. You Try #1: Graph 𝑓 𝑥 = 𝑥 2 −2𝑥+1 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

39. You Try #2: Graph 𝑓 𝑥 =−4 𝑥 2 +24𝑥−36 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

40. Do Now:
Take out homework packet and complete #6

41. So….
We know how to graph things that have one intercept or TWO x intercepts…. What about NO x intercepts?

42. Example Ten: Graph 𝑓 𝑥 = 𝑥 2 −4𝑥+8 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

43. Example Eleven: Graph 𝑓 𝑥 =− 𝑥 2 +5𝑥−7 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

44. Example Twelve: Graph 𝑓 𝑥 = 1 2 𝑥 2 +2𝑥+9 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

45. Example Thirteen: Graph 𝑓 𝑥 =−2 𝑥 2 −1 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

46. You Try #1: Graph 𝑓 𝑥 =− 𝑥 2 +4𝑥−5 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

47. You Try #2: Graph 𝑓 𝑥 =− 𝑥 2 +4𝑥−9 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

48. You Try #3: Graph 𝑓 𝑥 = 𝑥 2 +8𝑥+15 Direction of Opening
Vertex/Axis of Symmetry
Y intercept
X Intercepts
Domain
Range

49. Practice Problems Try some on your own/in your table groups
As always don’t hesitate to ask questions if you are confused OR ask your tablemates– they are your greatest resource!