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

c b 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 x-axis y-axis 1) 2) x-axis y-axis 3 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 x 2 + y 2 + 6x –2y + 9 = 0 if they exist. Solution: x-intercept(s): Set y = 0. x2 x2 + 6x + 9 = 0 ( x + 3) 2 = 0 x = - 3 Point ( -3,0) y-intercept(s): Set x = 0. y 2 –2y + 9 = 0 No Real Solutions No y-intercepts

Symmetries of Graphs of Equations in x and y TerminologyGraphical InterpretationTest 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 3 Complete the graph of the following if a) Symmetric w.r.t y-axis b) 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 4 Determine whether an equation is symmetric w.r.t y-axis, x-axis,origin or none a) y = 3x 4 + 5x 2 –4 b) y = -2x 5 +4x 3 +7x c) y = x3 x3 +x 2 Solution: a)Substitute x by –x y = 3( -x )4 )4 + 5 ( ) = 3x 4 + 5x 2 – 4 Same equation Substitute x by –x and y by - y (- y) = -2 (-x) 5 + 4( -x )3 )3 +7(-x) = 2x 5 – 4x 3 – 7x = - (-2x 5 +4x 3 +7 ) Same equation Substitute x by –x y = ( -x )5 )5 + ( )2)2 = - x5 x5 + x2x2 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 )2 + ( y – k )2 )2 = r2r2 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 )2 = 36. Solution: Center: C( 3, -5) Radius: r = 6 Example 6: Graph the above Circle (3,-5) (9, -5)(-3, -5) (3, -11) (3, 1 ) x y

Graphing Semi Circles Upper 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 x2 x2 + y2 y2 = 25. x2 x2 + y2 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. 1) Represents the upper half plane

Continued … To find right and left halves, we solve for x in terms of y ) Represents the right half plane 4) Represents the left half plane