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
- 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.
The graph initially has slope _____ in the region
The slope then becomes _____ for all between and
Then the slope changes to for all between and _____.
The minimum value of is _____, which is attained at
lines
(1) Initially, as increases to has a constant value of implying the slope is
(2) The slope of the graph for is
(3) The slope of the graph for is
(4) is minimized at where the value of isTherefore, our answer is
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 as a function of the number of sandwiches sold
Tom sells the sandwiches for $_____ each.
Tom will break even if he sells _____ sandwiches.
If Tom sells only sandwiches, his loss will be $_____.
To make $ in profit, Tom would have to sell _____ sandwiches.
Sales
(1) As increases by increases by Thus, each sandwich sells for $
(2) when Thus, Tom has to sell sandwiches to break even.
(3) He invested $ and the sales of sandwiches gives $ making the loss $
(4) He needs to sell more sandwiches from break-even point to make $ profit. So, he must sell a total of sandwiches.Therefore, our answer is
Amy and Ben run a race. The graph below shows the position in of each runner as a function of time in .
Amy runs at a constant speed of _____ during the first hour of the race.
Ben catches up with Amy _____ hours after the race started.
Amy takes a break for _____ hours in her hour race.
Ben eventually finishes the 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
- The two graphs intersect at implying that Ben catches up with Amy hours after the race started.
- The flat line in Amy's position graph continues from and to implying Amy took a break for hours in her hour race.
- The end points of Amy and Ben are and respectively. Thus, Ben finished the race hours ahead of Amy.
Therefore, our answer is
Fill in the blanks with appropriate numbers, based on the graph below.
The -intercept of the graph is _____.
The sum of the -intercepts of the graph is _____.
Consider the statement: " is a decreasing function for all " The minimum value of for the statement to be true is _____.
The maximum value of for occurs when _____.
graph2
(1) The graph meets with the -axis at
(2) The graph meets with the -axis at and which implies that the sum of the -intercepts is
(3) The graph shows that the function is increasing before and is decreasing after Thus the minimum value of that renders the statement true is -4.
(4) Since the function is increasing in the interval the maximum value of in this interval occurs whenTherefore, our answer is