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