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

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.

Example Does the point (-1,0) lie on the graph y = x3 – 1? No

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

Graphing a Quadratic Equation by Plotting Points Graph the equation: y=x²-5 x y=x²-5 (x,y) -2 -1 1 2

Graphing a Cubic Equation by Plotting Points Graph the equation: y=x³ x y=x³ (x,y) -2 -1 1 2

X and Y Intercepts An x–intercept 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.

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

What are the x and y intercepts of this graph given by the equation: y=x³-2x²-5x+6 x-intercepts: (-2,0)(1,0)(3,0) y-intercept: (0,6)

How do we find the x and y intercepts algebraically How do we find the x and y intercepts algebraically? First let’s examine the x-intercepts. For example: The graph to the right has the equation y=x²-6x+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=x²-6x+5 0=(x-5)(x-1) x-5=0 x-1=0 x=5,1 The x-intercepts are (1,0) and (5,0)

Next, let’s find the y-intercept. Equation: y=x²-6x+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=0²-6(0)+5 y=5 The y-intercept is (0,5)

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

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

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.

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.

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

Testing for Symmetry with respect to the x-axis Test the equation y²=x³ Solution: Replace y with –y (-y)²=x³ y²=x³ The equation is the same therefore it is symmetric with respect to the x-axis.

Testing from symmetry with respect to the y-axis Test the equation y²=x³ Solution: Replace x with –x y²=(-x)³ y²=-x³ The equation is NOT the same therefore it is NOT symmetric with respect to the y-axis.

Testing for Symmetry with respect to the origin Test the equation y²=x³ Solution: Replace x with –x and replace y with -y (-y)²=(-x)³ y²=-x³ The equation is NOT the same therefore it is NOT symmetric with respect to the origin.

Test for Symmetry: y = x5 + x Y-axis: x changes to –x Y = (-x)5 + -x y = -(x5 + x) No!

X-axis: y changes to –y -y = x5 + x y = -(x5 + x) No!

Origin: y changes to –y and x changes to –x -y = (-x)5 + -x -y = -(x5 + x) y = x5 + x Yes!