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