# 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 becomesfor 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 \(y=\)which is attained at \(x=4.\)_____,

(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 sellssandwiches._____

\(\quad (3)\) If Tom sells only \(6\) sandwiches, his loss will be $_____.

\(\quad (4)\) To make $\(10\) in profit, Tom would have to sellsandwiches._____

(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 Amyhours after the race started._____

\(\quad 3.\) Amy takes a break forhours in her \(7\) hour race._____

\(\quad 4.\) Ben eventually finishes the \(50 \text{ km}\) racehours ahead of Amy._____

- As the distance run is proportional to the time spent, the constant speed during the first hour for Amy is \(20 \text{ km/h}.\)
- The two graphs intersect at \(t=3.5,\) implying that Ben catches up with Amy \(3.5\) hours after the race started.
- 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.
- 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 \(x\) greater than or equal to \(a.\)" 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 \(x=\)_____.

(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 \[\left(-6,-8, -4, -5\right). \ _\square\]

**Cite as:**Graphs of Functions.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/interpreting-graphs-of-functions/