How To Test For Symmetry
Graphs and Symmetry
I. Homework
2. Symmetry (Geometry)
We say that a graph is symmetric with respect to the y axis if for every indicate (a,b) on the graph, there is likewise 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 examination for symmetry graphically using the graphing calculator.
We say that a graph is symmetric with respect to the ten axis if for every betoken (a,b) on the graph, there is also a indicate (a,-b) on the graph. Visually we have that the x axis acts as a mirror for the graph. Nosotros volition demonstrate several functions to test for symmetry graphically using the graphing estimator.
Nosotros say that a graph is symmetric with respect to the origin if for every point (a,b) on the graph, at that place is too 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, and then Q is also on the graph. We volition utilise the graphing calculator to examination for all iii symmetries.
4. Symmetry (Algebra)
To test algebraically if a graph is symmetric with respect the x axis, we replace all the y'south with -y and meet if we get an equivalent expression.
Examples:
A) For x - 2y = v nosotros replace with x - 2(-y) = 5. Simplifying we get
ten + 2y = v which is not equivalent to the original expression.
B) For x3 - y2 = 2 nosotros replace with tenthree - (-y)2 = two which is equivalent to the original expression, so that x3 - yii = 2 is symmetric with respect to the 10 centrality.
To test algebraically if a graph is symmetric with respect to the y axis, we supervene upon all the x's with -10 and see if we become an equivalent expression.
Example:
A) For y = 10two we supercede with y = (-ten)two = x2 so that y = xii is symmetric with respect to the y axis.
B) For y = ten3 we replace with y = (-x)3 = - x3 so that y = 103 is non 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 -ten and -y and meet if the result is equivalent to the original expression.
Examples:
A) For y = xthree, we replace with (-y) = (-x)3 so that -y = -xthree or y = xiii. Hence y = 103 is symmetric with respect to the origin.
B) For y = x2 we replace with -y = (-10)two and then that -y = x2 or y = -x2. Hence y = tenii is non symmetric with respect to the origin.
We volition do other examples in class as a grouping.
5. 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 10 axis, and then the y coordinate of the bespeak is 0. Hence to find the ten intercepts, we set y = 0 and solve.
Example: Notice the 10 intercepts of
y = 10ii + x - two
We prepare y = 0 so that
0 = 102 + x - 2 = (x + ii)(x - ane)
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 nosotros gear up x = 0 and detect y.
Example: Detect the y intercepts of y = x2 + ten - 2
Solution: We ready x = 0 to get: y = 0 + 0 - 2 = -2.
Hence the y intercept is at (0,-2).
How To Test For Symmetry,
Source: http://www.ltcconline.net/greenl/courses/103a/premid1/symm.htm
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