9.5. Solved problems

1. Find the intercepts of: 2y - 3x = 12

   y = 0, -3x = 12, x-intercept -4
   x = 0,  2y = 12, y-intercept  6
   

2. Find the intercepts of: 2x - 5y = 10

   y = 0,  2x = 10, x-intercept  5
   x = 0, -5y = 10, y-intercept -2
   

3. Find the intercepts of: 4y - 3x = 0

   y = 0, -3x = 0, x-intercept 0
   x = 0,  4y = 0, y-intercept 0
   

4. Find the intercepts of: y = 2

   Because the line is parallel to the x-axis, there is no x-intercept.
   The y-intercept is 2.
   

5. Find the distance and midpoints between: A{2, 4}, B{12, 28}

          _____________________
   DAB = √(12 - 2)2 + (28 - 4)2  = 26
   
   MAB = {(2 + 12)/2, (4 + 28)/2}  = {7, 16}
   

6. Write the equations for two circles that are tangent at the midpoint between their centers: A{0, 0}, B{6, 8}

          __________________
   DAB = √(6 - 0)2 + (8 - 0)2  = 10
   
   Each circle has radius of: DAB/2 = 5; then the two equations are:
   
   x2 + y2 = 25
   
   (x - 6)2 + (x - 8)2 = 25
   

7. Determine the symmetry of: x² + y² = 9

   y -> -y:  x2 + (-y)2 = 9 -> x2 + y2 = 9; 
   Circle has x-axis symmetry 
   
   x -> -x:  (-x)2 + y2 = 9 -> x2 + y2 = 9; 
   Circle has y-axis symmetry 
   
   x -> -x, y -> -y: (-x)2 + (-y)2 = 9 -> x2 + y2 = 9; 
   Circle has origin symmetry
   
   x -> y, y -> x: (y)2 + (x)2 = 9 -> x2 + y2 = 9; 
   Circle has symmetry about line y = x
   

8. Determine the center and radius of the circle: x² + y² + 8x + 2y + 8 = 0

   Complete the squares in x and y:
   (x2 + 8x + 16) - 16 + (y2 + 2y + 1) - 1 + 8 = 0
   (x + 4)2 + (y + 1)2 = 9
   Center:  r = 3, xo = -4, yo = -1
   

9. Sketch the graph of: 4x² + y² = 16

   First find the intercepts, 
   y = 0; 4x2 = 16 -> x2 = 4, x-intercepts: -2, 2
   x = 0; y2 = 16 y-intercepts: -4, 4
   Sketch the ellipse:
   
    
   
   

10. Determine the symmetry of: x² + 4y² = 4

    y -> -y: x2 + 4(-y)2 = 4 -> x2 + 4y2 = 4 
    Ellipse has x-axis symmetry.
    
    x -> -x: (-x)2 + 4y2 = 4 -> x2 + 4y2 = 4 
    Ellipse has y-axis symmetry.
    
    y -> -y, x -> -x: (-x)2 + 4(-y)2 = 4 -> x2 + 4y2 = 4 
    Ellipse has origin symmetry.
    
    y -> x, x -> y: (y)2 + 4(x)2 = 4 -> 4x2 + y2 = 4 
    Ellipse does not have symmetry with respect to line y = x.
    

11. Determine the symmetry of: x² - 4y² = 4

    y -> -y: x2 - 4(-y)2 = 4 -> x2 - 4y2 = 4 
    Hyperbola has x-axis symmetry.
    
    x -> -x: (-x)2 - 4y2 = 4 -> x2 - 4y2 = 4 
    Hyperbola has y-axis symmetry.
    
    y -> -y, x -> -x: (-x)2 - 4(-y)2 = 4 -> x2 - 4y2 = 4 
    Hyperbola has origin symmetry.
    
    y -> x, x -> y: (y)2 - 4(x)2 = 4 -> -4x2 + y2 = 4 
    Hyperbola does not have symmetry with respect to line y = x.
    

12. Find the intercepts of: y = x² + x - 6

    Factor the right hand side.
    y = (x + 3)(x - 2) 
    y = 0; 0 = (x + 3)(x - 2) -> x-intercepts: -3, 2
    x = 0; y = -6             -> y-intercept:  -6
    

13. Find the intercepts of: y = x² + 3x - 4

    Factor the right hand side.
    y = (x + 4)(x - 1) 
    
    y = 0; 0 = (x + 4)(x - 1) -> x-intercepts: -4, 1
    x = 0; y = -4             -> y-intercept:  -4
    

14. Find the intercepts of: y = x³ + 3x² + 3x + 1

    Factor the right hand side.
    y = (x + 1)3 
    
    y = 0; 0 = (x + 1)3 -> x-intercept: -1
    x = 0; y = 1        -> y-intercept: +1
    

15. Find the intercepts of: y = x² + 4

    Factor the right hand side.
    y = (x + 2i)(x - 2i)
    
    y = 0; 0 = (x + 2i)(x - 2i) -> no x-intercepts
    x = 0; y = 4                -> y-intercept: +4
    

16. Find the intercepts of:

         _______
    y = √ x + 9 
                _______
    y = 0; 0 = √ x + 9  -> x-intercept: -9
    x = 0; y = 3        -> y-intercept: +3
    

17. Determine the symmetry of: y = |x| + 2

    y -> -y; (-y) = |x| + 2 -> y = -|x| - 2 
    It is not symmetric with respect to the x-axis.
    
    x -> -x; y = |(-x)| + 2 -> y = |x| + 2
    It is symmetric with respect to the y-axis.
    
    y -> -y, x -> -x; (-y) = |(-x)| + 2 -> y = -|x| - 2
    It is not symmetric with respect to the origin.
    
    y -> x, x -> y; (x) = |(y)| + 2 -> x = |y| + 2
    It is not symmetric with respect to the line y = x.
    

18. Find the intercepts of: |y| = |x| + 2

    y = 0; 0 = |x| + 2 -> no x-intercept
    x = 0; |y| = 0 + 2 -> y-intercepts: -2, 2
    

19. Construct a table of ordered pairs for:

         ____
    y = √ 3x  + 1
    
    x      y
    ---------------
    0      1.000000
    1      2.732051
    2      3.449490
    3      4.000000
    4      4.464102
    5      4.872983
    

20. Construct a table of ordered pairs for:

         _______
    y = √ x + 3
    
     x      y
    ----------------
    -3      0.000000
    -2      1.000000
    -1      1.414214
     0      1.732051
     1      2.000000
     2      2.236068
     3      2.449490
     4      2.645751
     5      2.828427