Graphs and Symmetry

I.  Homework

II.  Symmetry (Geometry)

We say that a graph is symmetric with respect to the y axis if for every point (a,b) on the graph, there is also a point (-a,b) on the graph.  Visually we have that the y axis acts as a mirror for the graph.  We will demonstrate several functions to test for symmetry graphically using the graphing calculator.  

We say that a graph is symmetric with respect to the x axis if for every point (a,b) on the graph, there is also a point (a,-b) on the graph.  Visually we have that the x axis acts as a mirror for the graph.  We will demonstrate several functions to test for symmetry graphically using the graphing calculator.  

We say that a graph is symmetric with respect to the origin if for every point (a,b) on the graph, there is also a point (-a,-b) on the graph.  Visually we have that given a point P on the graph if we draw a line segment PQ through P and the origin such that the origin is the midpoint of PQ, then Q is also on the graph.  We will use the graphing calculator to test for all three symmetries.

IV.  Symmetry (Algebra)

To test algebraically if a graph is symmetric with respect the x axis, we replace all the y's with -y and see if we get an equivalent expression.  

Examples:  

A)  For x - 2y = 5 we replace with x - 2(-y) = 5.  Simplifying we get

x + 2y = 5 which is not equivalent to the original expression.

B)  For x³ - y² = 2 we replace with   x³ - (-y)² = 2 which is equivalent to the original expression, so that x³ - y² = 2 is symmetric with respect to the x axis.

To test algebraically if a graph is symmetric with respect to the y axis, we replace all the x's with -x and see if we get an equivalent expression.

Example:

A)  For y = x²  we replace with y = (-x)² =  x²  so that y = x²  is symmetric with respect to the y axis.

B)  For y = x³ we replace with  y = (-x)³ = - x³ so that y = x³ is not symmetric with respect to the y axis.

To test algebraically if a graph is symmetric with respect to the origin we replace both x and y with -x and -y and see if the result is equivalent to the original expression.  

Examples:  

A)  For y = x³, we replace with (-y) = (-x)³ so that -y = -x³ or y = x³.  Hence  y = x³  is symmetric with respect to the origin.  

B)  For y = x² we replace with -y =  (-x)² so that -y = x²  or y = -x².  Hence y = x² is not symmetric with respect to the origin.  

We will do other examples in class as a group.

V.  Intercepts

We define the x intercepts as the points on the graph where the graph crosses the x axis.  If a point is on the x axis, then the y coordinate of the point is 0.  Hence to find the x intercepts, we set y = 0 and solve.

Example:  Find the x intercepts of

y = x² + x - 2

We set y = 0 so that

0 =  x² + x - 2 = (x + 2)(x - 1)

Hence that x intercepts are at (-2,0) and (1,0)

We define the y intercepts of a graph to be the points where the graph crosses the y axis.  At these points the x coordinate is 0 hence to fine the y intercepts we set x = 0 and find y.

Example:   Find the y intercepts of  y = x² + x - 2

Solution:  We set x = 0 to get:  y = 0 + 0 - 2 = -2.  

Hence the y intercept is at (0,-2).