Graphs of Functions

Given a graph, how can we characterize its behavior? A graph may show different types of behavior in different regions and it may be useful to first break up the graph into sections to describe these different types of behavior. A few first properties to notice about the graph may include whether the graph is

• increasing or decreasing in a region
• positive, negative, or zero
• lies above or below a threshold in a region, or is constant in a region
• contains a given point $(x,y)$
• is a line segment in a region.

We can also use the graph to find the value of the minimum and maximum points lying in a region. Furthermore, if a graph within a region is a line, we can use any two points within this region to determine the slope of the line within the region.

Fill in the blanks with appropriate numbers, based on the graph below.

$\quad (1)$ The graph initially has slope _____ in the region $x \leq 2.$
$\quad (2)$ The slope then becomes _____ for all $x$ between $x=2$ and $x=4.$
$\quad (3)$ Then the slope changes to $1$ for all $x$ between $x=4$ and $x=$_____.
$\quad (4)$ The minimum value of $y$ is _____, which is attained at $x=4.$

lines

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(1) Initially, as $x$ increases to $2,$ $y$ has a constant value of $1,$ implying the slope is $0.$
(2) The slope of the graph for $2\le x\le 4$ is $\frac{(-3)-1}{4-2}=-2.$
(3) The slope of the graph for $4\le x\le 8$ is $\frac{1-(-3)}{8-4}=1.$
(4) $y$ is minimized at $x=4,$ where the value of $y$ is $-3.$

Therefore, our answer is $(0, -2, 8, -3).$ $_\square$

Tom bought bread, sliced cheese, and some vegetables for $20 in order to sell sandwiches at a soccer game. He made a total of 17 sandwiches, and the graph below shows the profit $P$ as a function of the number of sandwiches sold $n.$

$\quad (1)$ Tom sells the sandwiches for $_____ each.
$\quad (2)$ Tom will break even if he sells _____ sandwiches.
$\quad (3)$ If Tom sells only $6$ sandwiches, his loss will be $_____.
$\quad (4)$ To make $$10$ in profit, Tom would have to sell _____ sandwiches.

Sales

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(1) As $n$ increases by $1,$ $P$ increases by $2.$ Thus, each sandwich sells for $$2.$
(2) $P=0$ when $n=10.$ Thus, Tom has to sell $10$ sandwiches to break even.
(3) He invested $$20$ and the sales of $6$ sandwiches gives $$12,$ making the loss $$8.$
(4) He needs to sell $5$ more sandwiches from break-even point to make $$10$ profit. So, he must sell a total of $15$ sandwiches.

Therefore, our answer is $(2, 10, 8, 15).$ $_\square$

Amy and Ben run a $50 \text{ km}$ race. The graph below shows the position $($in $\text{km})$ of each runner as a function of time $($in $\text{h})$.

$\quad 1.$ Amy runs at a constant speed of _____ $\text{ km/h}$ during the first hour of the race.
$\quad 2.$ Ben catches up with Amy _____ hours after the race started.
$\quad 3.$ Amy takes a break for _____ hours in her $7$ hour race.
$\quad 4.$ Ben eventually finishes the $50 \text{ km}$ race _____ hours ahead of Amy.

Race

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1. As the distance run is proportional to the time spent, the constant speed during the first hour for Amy is $20 \text{ km/h}.$
2. The two graphs intersect at $t=3.5,$ implying that Ben catches up with Amy $3.5$ hours after the race started.
3. The flat line in Amy's position graph continues from $t=1$ and to $t=4,$ implying Amy took a break for $4-1=3$ hours in her $7$ hour race.
4. The end points of Amy and Ben are $(7, 50)$ and $(5, 50),$ respectively. Thus, Ben finished the $50 \text{ km}$ race $7-5=2$ hours ahead of Amy.

Therefore, our answer is $(20, 3.5, 3, 2).$ $_\square$

Fill in the blanks with appropriate numbers, based on the graph below.

$\quad (1)$ The $y$-intercept of the graph is _____.
$\quad (2)$ The sum of the $x$-intercepts of the graph is _____.
$\quad (3)$ Consider the statement: "$y$ is a decreasing function for all $x \ge a.$" The minimum value of $a$ for the statement to be true is _____.
$\quad (4)$ The maximum value of $y$ for $-9\le x\le -5$ occurs when $x=$_____.

graph2

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(1) The graph meets with the $y$-axis at $y=-6.$
(2) The graph meets with the $x$-axis at $x=-2$ and $x=-6,$ which implies that the sum of the $x$-intercepts is $(-2)+(-6)=-8.$
(3) The graph shows that the function is increasing before $x=-4,$ and is decreasing after $x=-4.$ Thus the minimum value of $a$ that renders the statement true is -4.
(4) Since the function is increasing in the interval $-9\le x\le -5,$ the maximum value of $y$ in this interval occurs when $x=-5.$

Therefore, our answer is $\left(-6,-8, -4, -5\right). \ _\square$