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Section 2.2

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.

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Section 2.2

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  1. 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

  5. Graphing a Quadratic Equation by Plotting Points Graph the equation: y=x²-5

  6. Graphing a Cubic Equation by Plotting Points Graph the equation: y=x³

  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. 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) How do we find the x and y intercepts algebraically? First let’s examine the x-intercepts.

  11. 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) Next, let’s find the y-intercept.

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

  13. 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. 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

  18. 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.

  19. 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.

  20. 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.

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

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

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

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