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Describing Key Features of Functions

Describing Key Features of Functions 1.4 - Solution

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a

We are asked to explore multiple properties of the graph. We will focus on one property at a time.

and intercepts

In the given graph the axis represents the time in hours starting at midnight and the axis the temperature in Let's mark the intercepts.

The graph intersects the axis at This means that the temperature at midnight was The graph also intersects the axis at about and This means that the temperature again was just after o'clock at night and at around half past in the morning.

Positive and Negative Parts

The function is positive when the graph is above the axis and negative when it is below.

Examining the graph we can determine at what hours the temperature was above and when it was below. Note that at and the temperature is neither positive nor negative. It is zero.

Increasing and decreasing parts

Regardless of whether the graph is positive or negative with respect to the axis, it is increasing when it's rising and decreasing when it's falling.

Examining the graph we can determine at what hours the temperature was increasing and when it was decreasing. Note that at and the graph is neither increasing nor decreasing.

Extrema

We can see the relative extrema for this given function by looking at the graph.

When the function goes from increasing to decreasing, like at , or from decreasing to increasing, like at we have a relative extrema. Here we have that the temperature had a local maximum at o'clock and a local minimum at o'clock.

End behavior

On the left end the arrow is pointing down. That indicates that the temperature was rising at midnight. On the right end we see that as increases, the value of increases as well. The end behavior of the graph indicates that at o'clock the temperature was increasing.


b

Let's study the properties of the graph one at the time.

and intercepts

Let's consider the given graph. The vertical axis represents the profit, in dollars, and the horizontal axis the number of television ads.

We see that the graph does not intersect the axis, so there is no intercept. Moreover, the function looks like it has intercept at This means that if they do not run any TV advertisement at all they make a profit of

Positive and negative parts

We can see from the graph that the given function is always positive. It means that they can run up to television ads and still the company makes a profit.

Increasing and decreasing parts

The graph shows how the profit increases as the number of television ads increases up to ads. Then, if more than ads are broadcast, the profit starts falling.

Extrema

The graph has its maximum, the relative extrema, at This means that to maximize the profit of the company, TV ads should be broadcast.

End behavior

Looking at the right part of the graph, as increases, the value of decreases. This means that buying more TV ads will lower the profit of the company.


c

Let's identify and interpret the various properties of the graph. We will begin with the intercepts.

and intercepts

The graph shows how high above the street level a descending elevator in Willis Tower is. The graph has one - and one -intercept.

The graph intersects the axis at This tells us that at time the elevator was meters above the street level. At the graph intersects the axis. That means that it took the elevator seconds to descend meters.

Positive and negative parts

In order to divide the function into positive and negative parts, we will use the -intercept we found earlier.

The function is positive for which means that the elevator is above street level, and negative for meaning that the elevator is in the basement.

Increasing and decreasing parts

From the graph we can see that the graph is decreasing for all Thus, the elevator is descending.

Extrema

The graph has no relative extrema.

End behavior

At the left end of the graph we see that as time decreases the elevation increases. That tells us that the elevator started its descent further up in the building. At the right end of the graph we see that as time increases the elevation decreases. Thus, when we stop monitoring the elevator's position it is going down.