Graphs and Models
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.
Graphing: Sketching a graph by plotting points. Example:
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:
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)
Example: Find x- and y-intercepts of the graph of
Symmetry: y-axis (x, y) (-x, y) x-axis (x, y) (x, -y) origin
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:
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.
Example:
Other examples: y-axis origin neither
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.
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.
Points of Intersection: To find points of intersection you "solve a system of equations". substitution elimination
Example: Find all points of intersection of the graphs of
Example: Find all points of intersection of the graphs of
Classwork: P.1: page 8 #1-4 all 19-41 odd 51-57 odd 63-69 odd (Don't sketch 41-57)