College Algebra Exercises Section 2.4

As you study these exercises, move your cursor over arrows, equal symbols, light bulbs, and highlighted words. Always check in with the wise old owl and his little apprentice by moving your cursor over them. Study with an active cursor.

 
The definition that pops up when you move the cursor over a highlighted word is what should pop into your mind when you read, hear, or speak that word. The material that pops up when you move the cursor over a light bulb is a suggested strategy for solving the problem. You should always formulate a similar strategy when you attempt to solve a problem. When you move the cursor over an explanation of a step in a solution, the property that pops up is the mathematical justification for taking that action. You must always be able to provide such a justification for every step you take in mathematics.

1. Find the f for which f(1) = 4 and f(0) = 6.
Solution:
Because we are looking for a linear function, the for the desired has the form f(x) = mx + b where m is the of the of f and b is its y-intercept.
The definition of graph of a function tells us that the points (1, f(1)) and (0, f(0)) are on the graph of the desired function.
The information in this problem tells us that that these two points are (1, 4) and (0, 6).
Clearly the point (0, 6) is the y-intercept and therefore b = 6.
That means the rule for the desired function has the form f(x) = mx + 6. (Note that we have the function)
The slope m can be determined by using the two points and the formula for :

The desired function is therefore the function f whose rule is f(x) = -2x + 6. (Note that now we have completely the function)

2. Find the f for which f(-3) = -8 and f(1) = 2.
Solution:
Because we are looking for a linear function, the for the desired has the form f(x) = mx + b where m is the of the of f and b is its y-intercept.
The definition of graph of a function tells us that the points (-3, f(-3)) and (1, f(1)) are on the graph of the desired function.
The information in this problem tells us that that these two points are (-3, -8) and (1, 2).
The slope m can be determined by using the two points and the formula for slope:

The rule for the desired function therefore has the form (Note that we have the function)
It is now possible to use two different methods to determine the rule for the desired function. Both methods will be presented. Using the point-slope equation works quite well when the desired function is linear. The other method, using the definition of graph, is applicable to all types of functions.
Using the Point-Slope Equation
Using the Definition of Graph
The , the newly calculated slope, and the point (1, 2) yields the following.

    (The function is completely .)
 The partially determined function ,
the definition of ,
and the fact that the point of the desired function. This yields:

   
Substitute this newly calculated value for b into the partially determined rule for the desired function to obtain:
     
as the rule for the desired function. (The function is completely .)

4. Find the f for which f(3) = 9 and f(-1) = -11.
Solution:
Because we are looking for a linear function, the for the desired has the form f(x) = mx + b where m is the of the of f and b is its y-intercept.
The definition of graph of a function tells us that the points (3, f(3)) and (-1, f(-1)) are on the graph of the desired function.
The information in this problem tells us that that these two points are (3, 9) and (-1, -11).
The slope m can be determined by using the two points and the formula for slope:

The rule for the desired function therefore has the form f(x) = 5x + b. (Note that we have the function)
It is now possible to use two different methods to determine the rule for the desired function. Both methods will be presented. Using the point-slope equation works quite well when the desired function is linear. The other method, using the definition of graph, is applicable to all types of functions.

Using the Point-Slope Equation
Using the Definition of Graph
The , the newly calculated slope, and the point (3, 9) yields the following.

   (The function is completely .)

The partially determined function f(x) = 5x + b,
the definition of ,
and the fact that the point of the desired function. This yields:

   9 = f(3) = 5(3) + b
   9 = 15 + b
    b = -6
Substitute this newly calculated value for b into the partially determined rule for the desired function to obtain:
     f(x) = 5x - 6.
as the rule for the desired function. (The function is completely .)

8. Find the f for which and f(-4) = -11.
Solution:
Because we are looking for a linear function, the for the desired has the form f(x) = mx + b where m is the of the of f and b is its y-intercept.
The definition of graph of a function tells us that the points and (-4, f(-4)) are on the graph of the desired function.
The information in this problem tells us that that these two points are and (-4, -11).
The slope m can be determined by using the two points and the formula for slope:

The rule for the desired function therefore has the form . (Note that we have the function)
It is now possible to use two different methods to determine the rule for the desired function. Both methods will be presented. Using the point-slope equation works quite well when the desired function is linear. The other method, using the definition of graph, is applicable to all types of functions.

Using the Point-Slope Equation
Using the Definition of Graph
The , the newly calculated slope, and the point (-4, -11) yields the following.

   (The function is completely .)

The partially determined function ,
the definition of ,
and the fact that the point of the desired function. This yields:

Substitute this newly calculated value for b into the partially determined rule for the desired function to obtain:
     .
as the rule for the desired function. (The function is completely .)

10. Sketch the graph of the function whose rule is given by
Solution:
Discussion
Graph -- Computer Generated
The y-intercept is .
The x-intercept is found by setting f(x) = 0 and solving for x.
The x-intercept for this function is .


14. Sketch the graph of the function whose rule is given by f(x) = -x2 + 8x
Solution:
Discussion
Graph -- Computer Generated

The graph of f(x) = -x2 + 8x = -x(x - 8) is a parabola which opens down..
The x-intercepts are found by solving the equation resulting from
f(x) = 0.
In this case we must solve the equation -x(x - 8) = 0.
From which we conclude the x-intercepts of f are 0 and 8.
The vertex is


16. Sketch the graph of the function whose rule is given by f(x) = x2 - 6x - 16
Solution:
Discussion
Graph -- Computer Generated
The graph of f(x) = x2 - 6x - 16 = (x - 8)(x + 2) is a parabola which opens up.
The x-intercepts are found by solving the equation resulting from
f(x) = 0.
In this case we must solve the equation (x - 8)(x + 2) = 0.
From which we conclude the x-intercepts of f are -2 and 8.
The vertex is


18. Sketch the graph of the function whose rule is given by f(x) = 8 - x3
Solution:
Discussion
Graph -- Computer Generated

The graph of f(x) = 8 - x3 = (2 - x)(4 + 2x + x2) is a smooth continuous curve with no sharp corners. It will try to cross the x-axis 3 times.
The x-intercepts are found by solving the equation resulting from
f(x) = 0.
In this case we must solve the equation (2 - x)(4 + 2x + x2) = 0.
From the Zero Factor Property we conclude that we must solve both
2 - x = 0 and x2 + 2x + 4 = 0. Notice the discriminant of the quadratic is negative so the solutions to the second equation are complex numbers and cannot be x-intercepts. The linear equation yields the
x-intercept 2.

This is not enough information to sketch a meaningful graph of the function. However, we can view this function as two simple transformations of the cubing function (in the library of basic functions) whose rule is h(x) = x3.

First note that the function k whose rule is k(x) = -x3 simply replaces each range value of h(x) = x3 with its opposite. That causes the graph to be rotated around the x-axis. Compare their graphs shown at the right (Figure 2-4-18a)on the same coordinate system.

Secondly note that the function f we are dealing with adds 8 to each range element of the function k. When every range value is increased by the same number, the affect is to raise the graph that many units. This is illustrated in Figure 2-4-18b at the right. Note that the distance between the red and green graphs, when measured along a vertical line is always 8 units.

The graph of the desired function whose rule is
f(x) = 8 - x3 = (2 - x)(4 + 2x + x2) shown in Figure 2-4-18c.

In summary, start by considering h(x) = x3, then note that k(x) = - x3 is a rotation around the x-axis and finally observe that the addition of 8 to get f(x) = 8 - x3 raises the graph 8 units.