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194

Chapter Five TRANSFORMATIONS FUNCTIONS OF AND THEIRGRAPHS

5.1

VERTICAL ANDHORIZONTAL HIFTS S

--

'1M

_n"rr ,""----

Suppose we shift the graph of some function vertically or horizontally, giving the graph of a new function. In this section we investigate the relationship between the formulas for the original function and the new function.

VerticalandHorizontal hift:TheHeating S Schedule Foran OfficeBuildingWe start with an example of a vertical shift in the context of the heating schedule for a building.

Example 1

To save money, an office building is kept warm only during business hours. Figure 5.1 shows the temperature, H, in of, as a function of time, t, in hours after midnight. At midnight (t = 0), the building's temperature is 50F. This temperature is maintained until 4 am. Then the building begins to warm up so that by 8 am the temperature is 70F. At 4 pm the building begins to cool. By 8 pm, the temperature is again 50F. Suppose that the building's superintendent decides to keep the building 5F warmer than before. Sketch a graph of the resulting function.H, temperatureCF) 75 H = p(t): New, vertically hifted s schedule

H, temperatureOF) (

70 60 50L

70 65 60

/

55 r-

'"H = f(t): Original eating h

H = f(t)t, time(hoursafter midnight)

50

scheduleI I I I

4

8

12

16

20

t, time (hours

.

24

4

8

12

16

20

24 aftermidnight)

Figure5.1: The heating schedule at an office building

Figure 5.2: Graph of new heating schedule, H = p( t), obtained by shifting original graph, H = f(t), upward by 5 units

Solution

The graph of f, the heating schedule function of Figure 5.1, is shifted upward by 5 units. The new heating schedule, H = p(t), is graphed in Figure 5.2. The building's overnight temperature is now 55F instead of 50F and its daytime temperature is 75F instead of 70F. The 5F increase in temperature corresponds to the 5-unit vertical shift in the graph.

The next example involves shifting a graph horizontally.

Example 2

The superintendent then changes the original heating schedule to start two hours earlier. The building now begins to warm at 2 am instead of 4 am, reaches 70F at 6 am instead of 8 am, begins cooling off at 2 pm instead of 4 pm, and returns to 50F at 6 pm instead of 8 pm. How are these changes reflected in the graph of the heating schedule?

5.1 VERTICALAND HORIZONTAL SHIFTS

195

Solution

Figure 5.3 gives a graph of H = q(t), the new heating schedule, which is obtained by shifting the graph of the original heating schedule, H = f (t), two units to the left.

70 60 50/

'"

'I

I

I

I

I

I

I

I

I

I

I

4

8

12

16

20

I t, time(hours 24 aftermidnight)

Figure 5.3: Graph of new heating schedule, H = q(t), found by shifting, j, the original graph 2 units to the left

Notice that the upward shift in Example 1 results in a warmer temperature, whereas the leftward shift in Example 2 results in an earlier schedule.

Formulas a Verticalor Horizontal hift for SHow does a horizontal or vertical shift of a function's graph affect its formula? Example 3 Solution In Example I, the graph of the original heating schedule, H = f (t), was shifted upward by 5 units; the result was the warmer schedule H = p(t). How are the formulas for f(t) and p(t) related? The temperature under the new schedule, p(t), is always 5F warmer than the temperature under the old schedule, f (t). Thus, New temperature at time t Writing this algebraically:~ New temperature at time t

Old temperature at time t

+5.

p(t)

~ Old temperature at time t

f(t)

+

5.

The relationship between the formulas for p and f is given by the equation p(t) = f(t) + 5. We can get information from the relationship p(t) = f(t) + 5, although we do not have an explicit formula for f or p. Suppose we need to know the temperature at 6 am under the schedule p(t). The graph of f (t) shows that under the old schedule f(6) = 60. Substituting t = 6 into the equation relating f and p gives p(6):..

p(6) = f(6) + 5 = 60+5 = 65. Thus,at 6 amthe temperature underthe newscheduleis 65F.

196

Chapter Five TRANSFORMATIONS FUNCTIONS OF AND THEIRGRAPHS

Example4

In Example 2 the heating schedule was changed to 2 hours earlier, shifting the graph horizontally 2 units to the left. Find a formula for q, this new schedule, in terms of f, the original schedule.The old schedule always reaches a given temperature 2 hours after the new schedule. For example, at 4 am the temperature under the new schedule reaches 60. The temperature under the old schedule reaches 60 at 6 am, 2 hours later. The temperature reaches 65 at 5 am under the new schedul\.:, but not until 7 am, under the old schedule. In general, we see that Temperature under new schedule at time t Algebraically, we have Temperature under old schedule at time (t + 2), two hours later.

Solution

q(t) = f(t + 2).This is a formula for q in terms of f.

Let's check the formula from Example 4 by using it to calculate q(14), the temperature under the new schedule at 2 pm. The formula gives

q(14) = f(14 + 2) = f(16). Figure 5.1 shows that f(16) = 70. Thus, q(14) = 70. This agrees with Figure 5.3.

Translations f a Function o andIts GraphIn the heating schedule example, the function representing a warmer schedule,

p(t) = f(t) + 5,has a graph which is a vertically shifted version of the graph of f. On the other hand, the earlier schedule is represented by

q(t) = f(t + 2)and its graph is a horizontally shifted version of the graph of f. Adding 5 to the temperature, or output value, f (t), shifted its graph up five units. Adding 2 to the time, or input value, t, shifted its graph to the left two units. Generalizing these observations to any function g:

If y = 9(x) is a function and k is a constant, then the graph of . y = g(x) + k is the graph of y = g(x) shifted vertically Ikl units. If k is positive, the shift is up; if k is negative, the shift is down.

. y = g(x

+ k) is the graph of y = g(x) shifted horizontally Ikl units. If k is positive, the

shift is to the left; if k is negative, the shift is to the right.

A vertical or horizontal shift of the graph of a function is called a translation because it does not change the shape of the graph, but simply translates it to another position in the plane. Shifts or translations are the simplest examples of transformations of a function. We will see others in later sections of Chapter 5.

5.1 VERTICALAND HORIZONTALSHIFTS

197

InsideandOutside Changes Since y = g(x + k) involves a change to the input value, x, it is called an inside change to g.Similarly, since y = g(x) + k involves a change to the output value, g(x), it is called an outside change. In general, an inside change in a function results in a horizontal change in its graph, whereas an outside change results in a vertical change. In this section, we consider changes to the input and output of a function. For the function

Q = f(t),a change inside the function's parentheses can be called an "inside change" and a change outside the function's parentheses can be called an "outside change."

Example 5 Solution

If n = f (A) gives the number of gallons of paint needed to cover a house of area A ft2, explain the meaning of the expressions f(A + 10) and f(A) + 10 in the context of painting. These two expressions are similar in that they both involve adding 10. However, for f(A + 10), the 10 is added on the inside, so 10 is added to the area, A. Thus,

n = f(A + 10) = '--v--' Area

Amount of paint needed to cover an area of (A + 10) ft2

Amount of paint needed to cover an area 10 ft2 larger than A.

The expressionf(A) + 10 representsan outsidechange.We are adding 10 to f(A), whichrepresents an amount of paint, not an area. We haveAmount of paint needed to cover region of area A

n = f(A)

~ Amount of paint

+ 10 =

+ 10gals=

IO gallons more paint than amount needed to cover area A.

In f(A + 10), we added 10 square feet on the inside of the function, which means that the area to be painted is now 10 ft2 larger. In f(A) + 10, we added 10 gallons to the outside, which means that we have 10 more gallons of paint than we need.

Example 6

Let 8(t) be the average weight (in pounds) of a baby at age t months. The weight, V, of a particular baby named Jonah is related to the average weight function 8(t) by the equation V = 8(t) + 2. Find Jonah's weight at ages t = 3 and t = 6 months. What can you say about Jonah's weight in general?

Solution

At t = 3 months, Jonah's weight is

V = 8(3) + 2.Since 8(3) is the average weight of a 3-month old boy, we see that at 3 months, Jonah weighs 2 pounds more than average. Similarly, at t = 6 months we have

V = 8(6)+ 2,

. 2 pounds .,

which means that, at 6 months, Jonah weighs 2 pounds more than average. In general, Jonah weighs more than average for babies of his age. ~

198

Chapter Five TRANSFORMATIONS FUNCTIONS OF AND THEIRGRAPHS

Example7

The weight, W, of another baby named Ben is related to s(t) by the equation W

= s(t + 4).

What can you say about Ben's weight at age t = 3 months? At t = 6 months? Assuming that babies increase in weight over the first year of life, decide if Ben is of average weight for his age,