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Section 2.2 Graphing Equations: Point-Plotting, Intercepts, and Symmetry

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Graphing Equations by Plotting Points The graph of an equation in two variables, x and y, consists of all the points in the xy plane whose coordinates (x,y) satisfy the equation.

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Example Does the point (-1,0) lie on the graph y = x 3 1? No

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Graphing an Equation of a Line by Plotting Points Graph the equation: y = 2x-1 xy=2x-1(x,y) -2 0 1 2

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Graphing a Quadratic Equation by Plotting Points Graph the equation: y=x-5 xy=x-5(x,y) -2 0 1 2

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Graphing a Cubic Equation by Plotting Points Graph the equation: y=x xy=x(x,y) -2 0 1 2

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X and Y Intercepts An xintercept of a graph is a point where the graph intersects the x- axis. A y-intercept of a graph is a point where the graph intersects the y- axis.

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Find the x and y intercepts. x-intercepts: (1,0) (5,0) y-intercept: (0,5)

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What are the x and y intercepts of this graph given by the equation: y=x-2x-5x+6 x-intercepts: (-2,0)(1,0)(3,0) y-intercept: (0,6)

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How do we find the x and y intercepts algebraically? First lets examine the x-intercepts. For example: The graph to the right has the equation y=x-6x+5. What is the y-coordinate for both x-intercepts? Zero. So to find x intercepts we can plug in zero for y and solve for x: 0=x-6x+5 0=(x-5)(x-1) x-5=0 x-1=0 x=5,1 The x-intercepts are (1,0) and (5,0)

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Next, lets find the y-intercept. Equation: y=x-6x+5. What is the x-coordinate for the y-intercept? Zero. So to find the y-intercept we can plug in zero for x and solve for y: y=0-6(0)+5 y=5 The y-intercept is (0,5)

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Symmetry The word symmetry conveys balance. Our graphs can be symmetric with respect to the x-axis, y-axis and origin.

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This graph is symmetric with respect to the x-axis. Notice the coordinates: (2,1) and (2,-1). The y values are opposite.

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This graph is symmetric with respect to the y-axis. What do you notice about the coordinates of this graph? The x values are opposite.

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This graph is symmetric with respect to the origin. What do you notice about the coordinates (2,3) and (-2,-3)? Both the x values and y values are opposite.

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Summary If a graph is symmetric about the X-axis, the y values are opposite Y-axis, the x values are opposite Origin, both the x and y values are opposites

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Testing for Symmetry with respect to the x-axis Test the equation y=x Solution: Replace y with y (-y)=x y=x The equation is the same therefore it is symmetric with respect to the x-axis.

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Testing from symmetry with respect to the y-axis Test the equation y=x Solution: Replace x with x y=(-x) y=-x The equation is NOT the same therefore it is NOT symmetric with respect to the y- axis.

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Testing for Symmetry with respect to the origin Test the equation y=x Solution: Replace x with x and replace y with -y (-y)=(-x) y=-x The equation is NOT the same therefore it is NOT symmetric with respect to the origin.

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Test for Symmetry: y = x 5 + x Y-axis: x changes to x Y = (-x) 5 + -x y = -(x 5 + x) No!

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X-axis: y changes to y -y = x 5 + x y = -(x 5 + x) No!

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Origin: y changes to y and x changes to x -y = (-x) 5 + -x -y = -(x 5 + x) y = x 5 + x Yes!