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

In the given graph the $x-$axis represents the time in hours starting at midnight and the $y-$axis the temperature in $_{∘}C.$ Let's mark the intercepts.

The graph intersects the $x-$axis at $(0,0).$ This means that the temperature at midnight was $0_{∘}C.$ The graph also intersects the $x-$axis at about $(3.2,0)$ and $(4.5,0).$ This means that the temperature again was $0_{∘}C$ just after $3$ o'clock at night and at around half past $4$ in the morning.

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

Examining the graph we can determine at what hours the temperature was above $0_{∘}C$ and when it was below. $Positive:Negative: 0<h<3.2andh>4.53.2<h<4.5 $ Note that at $x=0,$ $x=3.2,$ and $x=4.5,$ the temperature is neither positive nor negative. It is zero.

Regardless of whether the graph is positive or negative with respect to the $y-$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. $Increasing:Decreasing: 0≤h<2andh>42<h<4 $ Note that at $x=2$ and $x=4,$ the graph is neither increasing nor decreasing.

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 $h=2$, or from decreasing to increasing, like at $h=4,$ we have a relative extrema. Here we have that the temperature had a local maximum at $2$ o'clock and a local minimum at $4$ o'clock.

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 $x$ increases, the value of $y$ increases as well. The end behavior of the graph indicates that at $7$ o'clock the temperature was increasing.

b

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

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 $x-$axis, so there is no $x-$intercept. Moreover, the function looks like it has $y-$intercept at $20000.$ This means that if they do not run any TV advertisement at all they make a profit of $$20000.$

We can see from the graph that the given function is always positive. It means that they can run up to $32$ television ads and still the company makes a profit.

The graph shows how the profit increases as the number of television ads increases up to $16$ ads. Then, if more than $16$ ads are broadcast, the profit starts falling.

The graph has its maximum, the relative extrema, at $x=16.$ This means that to maximize the profit of the company, $16$ TV ads should be broadcast.

Looking at the right part of the graph, as $x$ increases, the value of $y$ 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.

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

The graph intersects the $y-$axis at $(0,45).$ This tells us that at time $t=0$ the elevator was $45$ meters above the street level. At $(5.5,0)$ the graph intersects the $x-$axis. That means that it took the elevator $5.5$ seconds to descend $45$ meters.

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

The function is positive for $x<5.5,$ which means that the elevator is above street level, and negative for $x>5.5,$ meaning that the elevator is in the basement.

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

The graph has no relative extrema.

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.