1. Graphs and Models

2. Objectives:
Sketch the graph of an equation both manually and via use of a calculator.
Find the intercepts of a graph.
Test a graph for symmetry with respect to an axis and the origin.
Find the points of intersections of two graphs.
Interpret mathematical models for real-life data.

3. Graphing:
Sketching a graph by plotting points.
Example:

4. Graphing Calculator: Zoom features: Table settings: Window settings:
Zoom 6 (non-trig functions)
Zoom 7 (trig functions)
Zoom 0 (zoom fit)
Table settings:
Window settings:

5. Intercepts: x-intercepts: Points where the graph crosses the x-axis.
Found by letting y=0 and solving for x.
y-intercepts:
Points where the graph crosses the y-axis. Found by letting x=0 and solving for y.
Calculator
Calculate value when x=0 (y-intercept)
Calculate Zero (x-intercept)

6. Example:
Find x- and y-intercepts of the graph of

7. Symmetry: y-axis (x, y) (-x, y) x-axis (x, y) (x, -y) origin

8. Testing for Symmetry:
The following tests show symmetry if they yield an equation equivalent to the original relation.
Replace x by (–x)  y-axis
Replace y by (–y)  x-axis
Replace x by (–x) and y by (–y)  origin
Example-Test the symmetry of:

9. Shortcut IF POLYNOMIALS:
All even powers of x 
symmetric to the y-axis.
All odd powers of x 
symmetric to the origin.
All even powers of y 
symmetric to x-axis.

10. Example:

11. Other examples:
y-axis
origin
neither

12. Example: We can use symmetry to help sketch a graph.
is symmetric to the x-axis.
all even powers of y, or
can replace y by –y
Can sketch upper part of graph by plotting points then reflect across the x-axis.

13. Graphing Calculator: How do you graph this on the calculator?
To graph on the calculator, you have to solve for y and then plot BOTH equations.

14. Points of Intersection:
To find points of intersection you "solve a system of equations".
substitution
elimination

15. Example:
Find all points of intersection of the graphs of

16. Example:
Find all points of intersection of the graphs of

17. Classwork:
P.1: page 8 #1-4 all odd odd odd (Don't sketch 41-57)