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

2. 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.

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

4. 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

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

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

7. 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.

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

9. 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)

10. 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)

11. 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)

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

13. 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.

14. 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.

15. 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.

16. 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

17. 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.

18. 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.

19. 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.

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

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

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