Math 5  Trigonometry  fall ’06  Chapter 3 Test (2)  Solutions

 

1.      Compute and simplify the average rate of change of  over the given interval.  Remember that the average rate of change on the interval [a, b] is the slope of the secant line connecting [a, f(a)] with  [b, f(b)].

a.        [0, 3]
SOLN:   

b.      [a, a+h]
SOLN:  

2.      Find the maximum value of the given function and state its range in interval notation.

a.      
SOLN:  The max is at (3, 8) and the range is  

b.     
SOLN:   has a max at (2, 9) and the range is .

3.      Consider the quadratic  

a.       Express the quadratic function in standard form.
SOLN:  

b.      Sketch its graph. 
SOLN:  The vertex is at .  The y-intercept is at (0,7) and the x-intercepts are  where  1.740 = 2.573 or 0.907   Putting this together we have a nice graph:

c.       What transformations would be required to transform this function to .
SOLN:  Shift 5/6 left, shift 109/12 down, reflect in the x-axis and compress vertically by 1/3.

 

4.      Given the graph of  shown at right, graph a.       SOLN:  This is a vertical stretch by a factor 2, so the point at (1,2) so stretched to (1,4), the point at (3,3) is stretch to (3,6), the point at (-1,-2) is stretched down to (-1,-4) and the point at (-3,-3) is stretched to (-3,-6) b.      SOLN:  This is a horizontal stretch by a factor of 2 so the point (1,2) is stretched to (2,2), the point at (3,3) is stretched to (6,3) and so on. c.       SOLN:  First shift 1 left, then reflect in y-axis, then stretch vertically by a factor 2.  Thus the point (1,2) goes to (0,2), stays there and is stretched to (0,4).  Similarly (3,3) goes to (2,3) then (-2,3) then (-2,6), and the other points, similarly, to produce the graph labeled at right. d.      This is a shift 1 to the left, followed by a reflection in the x axis and then a shift of 2 up. (1,2) goes to (0,2) then (0,-2) and then (0,0). The other points are similarly transformed to produce the graph labeled at right.

 

5.      A mouse stands at point A on the bank of a straight canal, 20 feet wide.  To reach point B, 70 feet down the canal on the opposite bank, it swims to a point P on the opposite bank and then crawls the remaining distance to B.  The mouse swims at 5 feet per minute and crawls at 10 feet per minute.  Model the total time of the mouse’s trip from A to B as a function of where he lands on the opposite side.

SOLN:  Let x = the distance down the shore the mouse lands on the other side (see diagram.)  Then the remaining distance it needs to travel is 70  x.  The distance D the mouse swims is the hypotenuse of a right triangle with legs of lengths 20 and x.  Thus .  Using the formula  we have the time swimming is  minutes and the time crawling is , so the total time is  

 

6.      Consider  

a.       Write a formula for the function that results from shifting 2 units left, reflecting
in the y-axis and then stretching horizontally by a factor 3, in that order.
SOLN:  

b.      What transformations on f(x), in order, would produce this formula:  
SOLN: Shift 1 unit right, stretch horizontally by 2, reflect in the x-axis and shift up 2.

7.      Suppose  and .

a.       Find the domain of
SOLN:  First, x must be in the domain of g, so x is not 2.  then the output of g must be greater than 1, so that f  is real-valued.  Thus the domain is the interval (2,3].

b.      Find the domain of  
SOLN:  First, x must be in the domain of f , so x ≥ 1.  Then the output of f must not be 2, so x is not 5.  Thus the domain is .

8.      Find a formula for the inverse function of  and sketch a graph for  and  together showing the symmetry through the line y = x.

SOLN:  so the inverse function is