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?

110 3

110

20

No

Graphing an Equation of a Line by Plotting Points

Graph the equation: y = 2x-1

x y=2x-1 (x,y)

-2

-1

0

1

2

Graphing a Quadratic Equation by Plotting Points

Graph the equation: y=x²-5

x y=x²-5 (x,y)

-2

-1

0

1

2

Graphing a Cubic Equation by Plotting Points

Graph the equation: y=x³

x y=x³ (x,y)

-2

-1

0

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? 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.

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

If the point (-3,2) is on a graph…

What point is also on the graph if it is symmetric to: x-axis

y-axis

origin

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!

y = x5 + x

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

y = x5 + x

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