MATH 1090 Algebra and Trigonometry
Problem Set 4 - Questions

1. The point P has coordinates (2, -1).   Find:
   1. the equation of the line of slope 4 that passes through P;
   2. the equation of the line perpendicular to the line in part (a) and passing through the point (2, 3); and
   3. the equation of the circle of radius 4 and centre at P.
   4. Sketch all three objects on the same diagram.

2. Sketch, on the same diagram, the curve and line whose equations are given by
   1. x² + y² + 6x - 8y = 0
   2. 4x - 3y + 24 = 0

3. For the function   y = x³ - x ,
   1. Sketch the curve from x = -2 to x = 2.
   2. Use the vertical line test to illustrate that   f (x)   is a function.
   3. Is   f (x)   even, odd or neither?
   4. Find the x and y intercepts.
   5. Find the domain and range of   f (x).
   6. Find the average rate of change of   f (x)   over each of the intervals
      1. x = 1   to   x = 2
      2. x = 1   to   x = 1.1
      3. x = 1   to   x = 1.01
   Challenge questions:
   7. Find the average rate of change of   f (x)   over the interval from   x = 1   to   x = 1 + h , (h ¹ 0).   Hence find the limit of the sequence of average rates of change that you found in part (f), as the second point approaches the first point at x = 1 .
   8. Generalize part (g) by finding and simplifying the difference quotient and find its limiting value as   h ® 0.
      [This is the instantaneous rate of change of f (x) at any point (x, f (x)) on the curve.]

4. Find the domain and range of the function  

5. Use transformations of one of the standard graphs on page 217 of the textbook,
               y = | x | ,       y = x² ,       y = Ö x ,
   to sketch the graph whose equation is
   1. y = 2 | x - 3 |
   2. 
   3. y = x² - 4 x + 5

6. Find the domain and inverse of the function and sketch the graph of   f   ^-1(x)   on top of the graph of   f (x):

Also:   Try the questions from exercise sets 2.1 to 2.6 of the textbook.
The solutions to this problem set will appear elsewhere on this web site.

[Index of Questions]

[Problem Set & Quiz Solutions]