section 2.2 graphing equations: point-plotting, intercepts, and symmetry
TRANSCRIPT
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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 = x3 – 1?
110 3
110
20
No
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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
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Graphing a Quadratic Equation by Plotting Points
Graph the equation: y=x²-5
x y=x²-5 (x,y)
-2
-1
0
1
2
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Graphing a Cubic Equation by Plotting Points
Graph the equation: y=x³
x y=x³ (x,y)
-2
-1
0
1
2
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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.
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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=x³-2x²-5x+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 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)
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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)
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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 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.
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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.
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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.
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Test for Symmetry: y = x5 + x
Y-axis: x changes to –x Y = (-x)5 + -x y = -(x5 + x) No!
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X-axis: y changes to –y -y = x5 + x y = -(x5 + x) No!
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Origin: y changes to –y and x changes to –x -y = (-x)5 + -x -y = -(x5 + x) y = x5 + x Yes!