Section 3.2 Graphs of Equations Objectives: Find the symmetries of equations with respect to x, y...
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Transcript of Section 3.2 Graphs of Equations Objectives: Find the symmetries of equations with respect to x, y...
Section 3.2
Graphs of Equations
Objectives:
•Find the symmetries of equations with respect to x, y axis and origin.
•Use the graphical interpretation
In this presentation I also show an introduction to x-intercepts and y-intercepts of an equation, graphically and algebraically as well as Circles
cb
Intercepts
Graphical Approach
x-axis
y-axis
a
d
a, b, and c are x-intercepts ( y = 0)
d is a y-intercept ( x = 0)
Algebraic Approach
x-intercept: Set y = 0 and solve for x
y-intercept: Set x = 0 and solve for y
Example 1
Find the x-intercept(s) and y-intercepts(s) if they exist.
-3
2
1.5 6 7
x-axis
y-axis1)
2)
x-axis
y-axis
3
-1 4
Solution:
x-intercept(s): x = -3, 1.5, 6 and 7
y-intercept(s): y = 2
Solution:
x-intercept(s): Does Not Exist ( D N E )
y-intercept(s): y = 3
Example 2
Find the x-intercept(s) and y-intercept(s) of the equation
x2 + y2 + 6x –2y + 9 = 0 if they exist.
Solution:
x-intercept(s): Set y = 0.
x2 + 6x + 9 = 0
( x + 3)2 = 0
x = - 3 Point ( -3,0)
y-intercept(s): Set x = 0. y2 –2y + 9 = 0
032914442 acb
No Real Solutions No y-intercepts
Symmetries of Graphs of Equations in x and y
Terminology Graphical Interpretation Test for symmetry
The graph is symmetric with respect to y-axis
(1) Substitution of –x for x leads to the same equation
The graph is symmetric with respect to x-axis
(2) Substitution of –y for y leads to the same equation
The graph is symmetric with respect to origin
(3) Substitution of –x for x and Substitution of –y for y
leads to the same equation
(x,y)(-x,y)
(x,y)
(x,-y)
(x,y)
(-x,-y)
Continue… Example 3Complete the graph of the following if
a) Symmetric w.r.t y-axisb) Symmetric w.r.t origin
c) Symmetry with respect to y-axis d) Symmetry with respect to origin
e) Symmetric w.r.t x-axis
Example 4Determine whether an equation is symmetric w.r.t y-axis, x-axis ,origin or none
a) y = 3x4 + 5x2 –4 b) y = -2x5 +4x3 +7x c) y = x3 +x2
Solution:a) Substitute x by –x
y = 3( -x )4 + 5 ( -x )2 - 4
= 3x4 + 5x2 – 4
Same equation
Substitute x by –x and y by - y
(-y) = -2 (-x)5 + 4( -x )3 +7(-x)
= 2x5 – 4x3 – 7x
= - (-2x5 +4x3 +7 )
Same equation
Substitute x by –x
y = ( -x )5 + ( -x )2
= - x5 + x2
Different equation
Symmetry w.r.t y-axis Symmetry w.r.t origin
Even if we substitute –y for y, we get different equations
Circles
Equation of a circle: ( x – h )2 + ( y – k )2 = r2
Center of the circle: C( h, k )
Radius of the circle: r
Diameter of the circle: d = 2r
rr
d= 2r
Example 5: Find the center and the radius of a circle whose equation is
( x – 3)2 + ( y + 5 )2 = 36.
Solution:
Center: C( 3, -5)
Radius: r = 6
Example 6: Graph the above Circle.6
66
6 (3,-5)(9, -5)(-3, -5)
(3, -11)
(3, 1 )
x
y
Graphing Semi CirclesUpper half, Lower half, right half, and left half
Let us find the equations of the upper half, lower half, right half and left half of the circle x 2 + y2 = 25.
x2 + y2 = 25 is a circle with center ( 0, 0 ) and radius r = 5. The graph of this circle is shown below.
To find upper and lower halves, we solve for y in terms of x.
2522 yx22 25 xy
225 xy
025 2 xy1) Represents the upper half plane
025 2 xy
-5 5
5025 2 xy
5-5
-5
5
0
-5
5-5
025 2 xy
Continued…
To find right and left halves, we solve for x in terms of y.
-5
5
5 025 2 yx
025 2 yx
3) Represents the right half plane
025 2 yx 4) Represents the left half plane
025 2 yx