1. AP Calculus Mrs. Mongold
P.1 Graphs and Models
AP Calculus
Mrs. Mongold

2. The Graph of an Equation
Use a graphing calculator if permitted, faster, easier, more accurate
If graphing calculator is not permitted, make a table of points and ALWAYS find a minimum of 5 points.

3. Intercepts and Zeros Find any x and y intercepts if they exist
Remember x intercepts and zeros are the same points so don’t do more work than you need to.
I would encourage you to find 5 points that are not intercepts in addition to any intercepts that you find, especially if you aren’t sure what the graph is supposed to look like!

4. Symmetry of a Graph 1. Symmetry with respect to the y-axis
When (x,y) is a point on the graph, (-x,y) is also on the graph. (mirror across the y-axis)
2. Symmetry with respect to the x-axis
When (x, y) is a point on the graph and (x, -y) is also on the graph. (mirror across the x-axis)
3. Symmetry with respect to the origin
When (x, y) is a point on the graph and (-x, -y) is also on the graph. (graph unchanged by 1800 rotation)

5. Tests for Symmetry The graph of an equation in x and y …
1. is symmetric with respect to the y-axis if replacing x by –x yields an equivalent equation.
2. is symmetric with respect to the x-axis if if replacing y with – y yields an equivalent equation
3. symmetric with respect to the origin if replacing x with –x and y with –y yields an equivalent equation

6. Example 1
Test y = 2x3 - x for symmetry

7. Example 1
Test y = 2x3 - x for symmetry
About the y-axis

8. Example 1 Test y = 2x3 - x for symmetry About the y-axis
Take y = 2x3 – x and replace x with -x

9. Example 1 Test y = 2x3 - x for symmetry About the y-axis
Take y = 2x3 – x and replace x with –x
So y = 2(-x)3 – (-x) and simplify

10. Example 1 Test y = 2x3 - x for symmetry About the y-axis
Take y = 2x3 – x and replace x with –x
So y = 2(-x)3 – (-x) and simplify
Result y = -2x3 + x

11. Example 1 Test y = 2x3 - x for symmetry About the y-axis
Take y = 2x3 – x and replace x with –x
So y = 2(-x)3 – (-x) and simplify
Result y = -2x3 + x
Not an equivalent equation, so no symmetry about the y-axis

12. Example 1 Cont.. Test y = 2x3 - x for symmetry About the x-axis
Take y = 2x3 – x and replace y with –y
So - y = 2x3 – x and simplify
Result y = -2x3 + x
Not an equivalent equation, so no symmetry about the x-axis

13. Example 1 Cont.. Test y = 2x3 - x for symmetry About the origin
Take y = 2x3 – x and replace s with –x and y with –y
So - y = 2(-x)3 – (-x) and simplify
Result -y = -2x3 + x  y = 2x3 – x
This is an equivalent equation, so symmetry about the origin does exist.

14. Symmetry Tests and Graphing
In addition to your points like intercepts and zeros symmetry helps with graphing and saves you a ton of time which in AP world is precious!
Be smart when graphing without a graphing calculator

15. Points of Intersection
Sketch the equations in the same rectangular coordinate system, either by hand or on a graphing calculator
Use your calculator to find intercepts if permitted if not, find intercepts by hand. This is a skill that we use in AP Calculus a lot with integration! So get good at it and remember your Algebra II 

16. Mathematical Models
Again use your calculator if permitted, if not be as efficient as possible with your time!
Strive for accuracy and simplicity (although these concepts are often conflicting it is possible to achieve both in many cases)

17. Homework P.1
Every other Even