1 Graphs of basic functions (Section 2.6 , page 257)
2 Symmetry (Section 2.2 , page 188)
3 Completing the square (Section 1.3 , page 109)
Blood pressure is the force of blood per unit area against the walls of the arteries. Blood pressure is recorded as two numbers: the systolic pressure (as the heart beats) and the diastolic pressure (as the heart relaxes between beats). If your blood pressure is “120 over 80,” it is rising to a maximum of 120 millimeters of mercury as the heart beats and is falling to a minimum of 80 millimeters of mercury as the heart relaxes.
The most precise way to measure blood pressure is to place a small glass tube in an artery and let the blood flow out and up as high as the heart can pump it. That was the way Stephen Hales (1677–1761), the first investigator of blood pressure, learned that a horse’s heart could pump blood 8 feet 3 inches, up a tall tube. Such a test, however, consumed a great deal of blood. Jean Louis Marie Poiseuille (1797–1869) greatly improved the process by using a mercury-filled manometer, which allowed a smaller, narrower tube.
In Exercise 124, we investigate Poiseuille’s Law for arterial blood flow.
1 Define transformations of functions.
If a new function is formed by performing certain operations on a given function f, then the graph of the new function is called a transformation of the graph of f. For example, the graphs of y=|x|+2
2 Use vertical or horizontal shifts to graph functions.
A transformation that changes only the position of a graph but not its shape is called a rigid transformation. We first consider rigid transformations involving vertical shifts and horizontal shifts of a graph.
Let f(x)=|x|, g(x)=|x|+2,
Make a table of values and graph the equations y=f(x), y=g(x),
Notice that in Table 2.14 and Figure 2.78, each point (x, |x|)
x | y=|x| |
g(x)=|x|+2 |
h(x)=|x|−3 |
---|---|---|---|
−5 |
5 | 7 | 2 |
−3 |
3 | 5 | 0 |
−1 |
1 | 3 | −2 |
0 | 0 | 2 | −3 |
1 | 1 | 3 | −2 |
3 | 3 | 5 | 0 |
5 | 5 | 7 | 2 |
Let
Sketch the graphs of all three functions on the same coordinate plane. Describe how the graphs of g and h relate to the graph of f.
Example 1 illustrates the concept of the vertical (up or down) shift of a graph.
Next, we consider the operation that shifts a graph horizontally.
Let f(x)=x2, g(x)=(x−2)2,
First, notice that all three functions are squaring functions.
Each point (x, x2)
Each point (x, x2)
Let
Sketch the graphs of all three functions on the same coordinate plane. Describe how the graphs of g and h relate to the graph of f.
Sketch the graph of
3 Use reflections to graph functions.
We now consider rigid transformations that reflect a graph about a coordinate axis.
Consider the graph of f(x)=x2,
x | f(x)=x2 |
g(x)=−f(x)=−x2 |
---|---|---|
−3 |
9 | −9 |
−2 |
4 | −4 |
−1 |
1 | −1 |
0 | 0 | 0 |
1 | 1 | −1 |
2 | 4 | −4 |
3 | 9 | −9 |
To compare the graphs of f(x) and g(x)=f(−x),
x | f(x)=√x |
−x |
g(x)=√−x |
---|---|---|---|
−4 |
Undefined | 4 | 2 |
−1 |
Undefined | 1 | 1 |
0 | 0 | 0 | 0 |
1 | 1 | −1 |
Undefined |
4 | 2 | −4 |
Undefined |
Explain how the graph of y=−|x−2|+3
Start with the graph of y=|x|.
Step 1 Shift the graph of y=|x|
Step 2 Reflect the graph of y=|x−2|
Step 3 Finally, shift the graph of y=−|x−2|
Explain how the graph of y=−(x−1)2+2
Use the graph of f(x)=(x+1)2−4
The graph of f(x)=(x+1)2−4
We know that
This means that the portion of the graph on or above the x-axis (y≥0)
Use the graph of f(x)=2x−4
4 Use stretching or compressing to graph functions.
We now look at transformations that distort the shape of a graph, called nonrigid transformations. We consider the relationship of the graphs of y=af(x)
Let f(x)=|x|, g(x)=2|x|,
The graphs of y=|x|, y=2|x|,
x | f(x)=|x| |
g(x)=2|x| |
h(x)=12|x| |
---|---|---|---|
−2 |
2 | 4 | 1 |
−1 |
1 | 2 | 12 |
0 | 0 | 0 | 0 |
1 | 1 | 2 | 12 |
2 | 2 | 4 | 1 |
The graph of y=2|x|
The graph y=12|x|
Let f(x)=√x
Given a function y=f(x),
Notice that replacing the variable x with x±c,
Now consider the graphs of y=f(x)
The graph of the function y=f(x)
f(12x)
f(2x)
f(−2x)
Note that the domain of f is [−2, 2]
To graph y=f(12x),
To graph y=f(2x),
Therefore, we transform each point (x, y) in Figure 2.92 to (12x, y)
To graph y=f(−2x),
The graph of a function y=f(x)
f(12x)
f(2x)
5 Use a sequence of transformations to graph functions.
When graphing requires more than one transformation of a basic function, we perform transformations in the following order:
Horizontal shift
Stretch or compression
Reflection
Vertical shift
To graph y=af(bx−c)+d
Graph | Change each graph point (x, y) on the graph in the previous step to | |
Step 0 Original graph | y=f(x) |
(x, y) |
Step 1 Horizontal Shift | y=f(x−c) |
(x+c, y) |
Step 2 Stretch/Compression |
|
(x|b|, y) (x, |a|y) |
Step 3 Reflections |
|
(−x, y) (x,−y) |
Step 4 Vertical shift | y=af(bx−c)+d |
(x, y+d) |
Sketch the graph of the function f(x)=−2(x−1)2+3.
Begin with the graph of the basic function g(x)=x2.
Step 0
Step 1
Step 2
Step 3
Step 4
Sketch the graph of the function
Use the graph of y=f(x)
Use the graph of y=f(x)
The graph of y=f(x)−3
The graph of y=f(x+5)
The graph of y=f(bx)
The graph of y=f(−x)
True or False. The graph of y=f(x)
True or False. The graphs of y=f(x)
True or False. You get the same graph by shifting the graph of y=x2
True or False. Combining horizontal and vertical shifts of a graph preserves the shape of the original graph.
In Exercises 9–22, describe the transformations that produce the graphs of g and h from the graph of f.
f(x)=√x
g(x)=√x+2
h(x)=√x−1
f(x)=|x|
g(x)=|x|+1
h(x)=|x|−2
f(x)=x2
g(x)=(x+1)2
h(x)=(x−2)2
f(x)=1x
g(x)=1x+2
h(x)=1x−3
f(x)=√x
g(x)=√x+1−2
h(x)=√x−1+3
f(x)=x2
g(x)=−x2
h(x)=(−x)2
f(x)=|x|
g(x)=−|x|
h(x)=|−x|
f(x)=√x
g(x)=2√x
h(x)=√2x
f(x)=1x
g(x)=2x
h(x)=12x
f(x)=x3
g(x)=(x−2)3+1
h(x)=−(x+1)3+2
f(x)=√x
g(x)=−√x+1
h(x)=√−x+1
f(x)=〚x〛
g(x)=〚x−1〛+2
h(x)=3〚x〛−1
f(x)=3√x
g(x)=3√x+1
h(x)=3√x+1
f(x)=3√x
g(x)=23√1−x+4
h(x)=−3√x−1+3
In Exercises 23–34, match each function with its graph (a)–(l).
y=−|x|+1
y=−√−x
y=√x2
y=12|x|
y=√x+1
y=2|x|−3
y=1−2√x
y=−|x−1|+1
y=(x−1)2
y=−x2+3
y=−2(x−3)2−1
y=3−√1−x
In Exercises 35–62, graph each function by starting with a function from the library of functions and then using the techniques of shifting, compressing, stretching, and/or reflecting.
f(x)=x2−2
f(x)=x2+3
g(x)=√x+1
g(x)=√x−4
f(x)=|x|+2
f(x)=|x|−1
f(x)=x3+2
f(x)=x3−1
f(x)=1x+1
f(x)=1x−2
f(x)=(x−3)3
f(x)=(x+2)3
f(x)=√x−1
f(x)=√x+2
h(x)=|x+1|
h(x)=|x−2|
f(x)=(x+1)3
f(x)=(x−3)3
f(x)=1x−3
f(x)=1x+2
f(x)=√−x
f(x)=−√x
f(x)=−x2
f(x)=−x3
f(x)=2x2
f(x)=13x2
f(x)=2|x|
f(x)=13|x|
In Exercises 63–74, graph each function by starting with a function from the library of functions and then combining shifting and reflecting techniques.
f(x)=(x−2)2+1
f(x)=(x−3)2−5
f(x)=5−(x−3)2
f(x)=2−(x+1)2
f(x)=√x+1−3
f(x)=√x−2+1
f(x)=√1−x+2
f(x)=−√x+2−3
f(x)=|x−1|−2
f(x)=−|x+3|+1
f(x)=1x−1+3
f(x)=2−1x+2
In Exercises 75–82, graph each function by starting with a function from the library of functions and then combining shifting, compressing, stretching, and/or reflecting techniques.
f(x)=2(x+1)2−1
f(x)=13(x+1)2+2
f(x)=2−12(x−3)2
f(x)=1−3(x−3)2
f(x)=2√x+1−3
f(x)=√2x−2+1
f(x)=−2|x−1|+2
f(x)=−12|3−x|−1
In Exercises 83–94, write an equation for a function whose graph fits the given description.
The graph of f(x)=x3
The graph of f(x)=√x
The graph of f(x)=|x|
f(x)=√x
The graph of f(x)=x2
The graph of f(x)=x2
The graph of f(x)=√x
The graph of f(x)=√x
The graph of f(x)=x3
The graph of f(x)=x3
The graph of f(x)=|x|
The graph of f(x)=|x|
In Exercises 95–108, graph the function y=g(x)
g(x)=f(x)−1
g(x)=f(x)+3
g(x)=−f(x)
g(x)=f(−x)
g(x)=f(2x)
g(x)=f(12x)
g(x)=f(x+1)
g(x)=f(x−2)
g(x)=2f(x)
g(x)=12f(x)
g(x)=f(x−1)+2
g(x)=−f(−2x)−1
g(x)=f(1−2x)
g(x)=f(3−2x)
In Exercises 109–118, graph (a) y=g(x) and (b) y=|g(x)|, given the following graph of y=f(x).
g(x)=f(x)+1
g(x)=−2f(x)
g(x)=f(12x)
g(x)=f(−2x)
g(x)=f(x−1)
g(x)=f(2−x)
g(x)=−2 f(x+1)+3
g(x)=−f(−x+1)−2
g(x)=2f(1−2x)−3
g(x)=−f(−2−2x)+1
In Exercises 119–128, let f be the function that associates the employee number x of each employee of the ABC Corporation with his or her annual salary f(x) in dollars.
Across-the-board raise. Each employee was awarded an across-the-board raise of $800 per year. Write a function g(x) to describe the new salary.
Percentage raise. Suppose each employee was awarded a 5% raise. Write a function h(x) to describe the new salary.
Across-the-board and percentage raises. Suppose each employee was awarded a $500 across-the-board raise and an additional 2% of his or her increased salary. Write a function p(x) to describe the new salary.
Across-the-board percentage raise. Suppose the employees making $30,000 or more received a 2% raise, while those making less than $30,000 received a 10% raise. Write a piecewise function to describe these new salaries.
Health plan. The ABC Corporation pays for its employees’ health insurance at an annual cost (in dollars) given by
where x is the number of employees covered.
Use transformations of the graph of y=√x to sketch the graph of y=C(x).
Assuming that the company has 400 employees, find its annual outlay for the health coverage.
Poiseuille’s Law. For an artery of radius R, the velocity v (in mm/min) of blood flow at a distance r from the center of the artery is given by
where c is a constant.
In an artery for Angie, c=1000 and R=3 mm. Find the velocity of blood flow in this artery at
The center of the artery.
The inner linings of the artery.
Midway between the center and the inner linings.
Demand. The weekly demand for paper hats produced by Mythical Manufacturers is given by
where x represents the number of hats that can be sold at a price of p cents each.
Use transformations on the graph of y=p2 to sketch the graph of y=x(p).
Find the price at which 69,160 hats can be sold.
Find the price at which no hats can be sold.
Revenue. The weekly demand for cashmere sweaters produced by Wool Shop, Inc., is x(p)=−3p+600. The revenue is given by R(p)=−3p2+600p. Describe how to sketch the graph of y=R(p) by applying transformations to the graph of y=p2. [Hint: First, write R(p) in the form −3(p−h)2+k.]
Daylight. At 60° north latitude, the graph of y=f(t) gives the number of hours of daylight. (On the t-axis, note that 1=January and 12=December.)
Sketch the graph of y=f(t)−12.
Use the graph of y=f(t) of Exercise 127 to sketch the graph of y=24−f(t). Interpret the result.
In Exercises 129 and 130, the graphs of y=f(x) and y=g(x) are given. Find an equation for g(x) if the graph of g is obtained from the graph of f by a sequence of transformations.
In Exercises 131–138, by completing the square on each quadratic expression, use transformations on y=x2 to sketch the graph of y=f(x).
f(x)=x2+4x
[Hint: x2+4x=(x2+4x+4)−4=(x+2)2−4.]
f(x)=x2−6x
f(x)=−x2+2x
f(x)=−x2−2x
f(x)=2x2−4x
f(x)=2x2+6x+3.5
f(x)=−2x2−8x+3
f(x)=−2x2+2x−1
In Exercises 139–144, sketch the graph of each function.
y=|2x+3|
y=|〚x〛|
y=|4−x2|
y=|1x|
y=√|x|
y=〚|x|〛
The x-intercepts of the function f are −1, 0, 2. Find the corresponding x-intercepts for the following functions.
y=f(x+2)
y=f(x−2)
y=−f(x)
y=f(−x)
y=f(2x)
y=f(12x)
The y-intercept of the function f is 2. Find the corresponding y-intercepts for the following functions.
y=f(x)+2
y=f(x)−2
y=−f(x)
y=f(−x)
y=2f(x)
y=12f(x)
Let a function f have domain [−1, 3] and range [−2, 1]. Find the corresponding domain and range for the following functions.
y=f(x+2)
y=f(x)−2
y=−f(x)
y=f(−x)
y=2f(x)
y=12 f(x)
Let a function f have a relative maximum at x=1 and a relative minimum at x=2. Find the corresponding relative maxima and minima for the following functions.
y=f(x+2)
y=f(x)−2
y=−f(x)
y=f(−x)
y=2f(x)
y=12f(x)
In Exercises 149–154, perform the indicated operations.
(5x2+5x+7)+(x2+9x−4)
(x2+2x)+(6x3−2x+5)
(5x2+6x−2)−(3x2−9x+1)
(x3+2)−(2x3+5x−3)
(x−2)(x2+2x+4)
(x2+x+1)(x2−x+1)
In Exercises 155–158, find the domain of each function.
f(x)=2x−3x2−5x+6
f(x)=x−2x2−4
f(x)=√2x−3
f(x)=1√5−2x
In Exercises 159–162, solve the given inequality.
x−1x−10<0
−3(1−x)2>0
−2x+8x2+1≤0
(x−3)(x−1)(x−5)(x+1)≥0