1. GRAphing
By Cole Funk

2. Graphing in general
Graphing is a way to represent the relationship between two or more variables
The most powerful tool in your graphing toolbox is plotting!
Plotting is finding specific points on a graph by plugging numbers into the function or by collecting data and connecting the points to make the shape of the function
Some things to remember:
Go as far as you can on the graphs for the most accurate picture
Don’t forget your arrows if given the function
Don’t forget to make sure your line is solid or dashed based on the function
Don’t forget to shade regions of the graph if necessary
Don’t forget to plot all x and y intercepts!

3. General forms
It is extremely important in graphing to know the general form of a given type of an equation
A general forms of a line is
Standard form: y = mx + b
Two general forms of a quadratic are
Standard form: y = ax2 + bx + c
Vertex form: y = a(x - h)2 + k
A general form of a square root function is
y = (ax + b)1/2 + c

4. Solving for a function given points
I will give you guys points and tell you what type of function it is
The number of points I give you must be equal to the number of variables you need to solve for unless it’s a special case (which I will tell you about)
Use the general forms to know which variables you need to solve for (anything that isn’t x or y)
Steps:
Plug in points to make different equations (a system)
Solve the system
Plug coefficients and constants back into the general form to get the specific form of the function

5. Graphing lines Standard form: y = mx + b
m is the slope, b is the y-intercept (where x = 0)
Slope formula: m = (y2 – y1)/(x2 – x1)
Slope is rise over run! If you make the slope into a fraction, you know how far to go up and to the right or down and to the left. Negative slope means go up and to the left or down and to the right.

6. Graphing lines by plotting
Choose values for x and plug them in to the equation to get their respective y-values
Place points on the graph
Connect using a line

7. X Y

8. Practice problems Graph the following: y = 2x + 3 y = -x + 5

9. Solving for a line
Use the general form of a line to solve for the function given each set of points
(3, 2) , (4, 6)
(-5, 3) , (3, 5)
(7, 1) , (4, 5)

10. Graphing a quadratic using standard form
Standard form: y = ax2 + bx + c
Vertex: (-b/2a, f(-b/2a))
Remember, there is a line of symmetry that runs through the vertex vertically at x = -b/2a. This means the function is symmetrical on either side of it. You only have to plot one side of it to know the other side!
Try to plot all zeros (roots)
If a > 0, then it opens upward. If a < 0, then it opens downward

11. Graph each of the following: y = x2 + 2x + 1 y = x2 + 5x + 6
Practice problems
Graph each of the following:
y = x2 + 2x + 1
y = x2 + 5x + 6
y = -x2 + x + 2

12. Graphing a quadratic using Vertex form
Vertex form: y = a(x – h)2 + k
Vertex is at (h, k)
If a > 0, then it opens upward. If a < 0, then it opens downward

13. Practice problems
Graph the following:
y = 2(x – 3)2 + 2
y = 5(x + 2)2 – 1
y = -6(x + 8)2 + 5

14. Converting Vertex form to standard form
If there are two general forms for a quadratic, there must be a way to convert from one to the other!
Steps to convert from vertex form to standard form:
Distribute
Simplify

15. Practice problems
Convert to standard form:
y = 5(x – 4)2 - 1
y = -2(x + 1)2 - 6
y = 3(x – 2)2 + 2

16. Converting to vertex form from standard form
Remember that (x + b)2 = x2 + 2bx + b2
Vertex form: y = a(x – h)2 + k
Steps:
Starting from standard form, factor a out of the first two terms
Find what is next to x and divide it by 2 to get b (from the first equation on this slide)
Square what you just got to find what the last term would be
Take that out of c (from the standard form equation)
Use the first equation on this slide to factor the first three terms
Distribute a
Simplify

17. Convert to vertex form Y = x2 + 5x + 6 Y = 2x2 - 4x + 6
Practice problems
Convert to vertex form
Y = x2 + 5x + 6
Y = 2x2 - 4x + 6
Y = 3x2 + 9x - 18

18. Solving for a quadratic
To solve for a quadratic, I need to give you three points. I also MUST tell you that it’s a quadratic.
I want you to know how to solve for standard form. In my opinion, it is better to solve for vertex form for graphing purposes. However, it is usually much easier to solve for standard form.
General forms:
y = ax2 + bx + c
y = a(x – h)2 + k
Steps:
Plug in all points to the general equation of your choice
Solve the system by any means
Plug coefficients and constants in to the general equation you chose to get the specific form (your answer)

19. Practice Questions
Find the function for each set of points given that they’re quadratics.
(-2, 1), (-1, 0), (1, 4)
(-3, 0), (-1, 2), (0, 6)
(4, 6), (2, -4), (1, -6)