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

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.

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!

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)

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

Example 1 Test y = 2x3 - x for symmetry

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

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

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

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

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

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

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.

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

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 

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)

Homework P.1 Every other Even