Graphs and Symmetry

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Graphs and Symmetry

Symmetry (Geometry)

Definition

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.

$Graph that is pretty flat near x=0 and goes up on the left and the right. It is symmetric about the y-axis.$

Definition

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.

Definition

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.

$Graph of a curve that goes up and then down in the second quadrant and down and up in the fourth quadrant. Points P and Q are on the graph with the line connecting them going through the origin.$

We will use the graphing calculator to test for all three symmetries.

Symmetry (Algebra)

Examples:
1. x-axis Symmetry
  
  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.
  1. 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.
  2. 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.
2. y-axis Symmetry
  
  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.
  1. For
    
            y = x²
    
    we replace with
    
            y = (-x)² = x²
    
    so that
    
            y = x²
    
    is symmetric with respect to the y axis.
  2. For
    
            y = x³
    
            we replace with
    
            y = (-x)³ = - x³
    
    so that
    
            y = x³
    
            is not symmetric with respect to the y axis.
3. Origin Symmetry
  
  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.
  1. 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.
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).

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