Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.
Learn how to sketch the graph of a function based on a description of its behavior. You can do this even if you don't know any exact numbers for the input or output values!
You learned in another lesson about the way we graph functions: the input is shown on the x-axis with the output on the y-axis. So if you have a particular written function to graph, you can do it by plugging in different sets of inputs to get input-output pairs and then graphing those as points and connecting the lines.
In this lesson, we're doing something different. We're not going to be looking at specific points. Instead, we're going to be looking at the general pattern of behavior of a given function. Is it linear or not? Do x and y increase together, or does one increase as the other decreases? Is the function increasing or decreasing at a constant rate or not?
This is called its qualitative features, patterns or trends in behavior. In this lesson, you'll learn to draw a graph showing the qualitative features of a function, even if you can't plot any points.
Graphing Function Behavior
To map the basic qualitative features of a function onto the graph, you don't have to know specific numbers, but you do need to know how certain patterns of behavior look on a graph. Here are some of the most common:
Increase vs. decrease: When a function is increasing, it means that f(x), or y, gets bigger as x gets bigger. If a function is increasing, the graph slopes upward from left to right. If the function is decreasing, it means that f(x), or y, gets smaller as x gets bigger. In this case, the graph slopes downward.
Rate of increase or decrease: The rate of change refers to how quickly f(x), or y, changes. The faster the rate of change, the steeper the slope of the function when it's shown as a graph. So if a function is increasing at a fast rate, the slope will be very steep. If it's increasing at a slow rate, the slope will be gentler.
If the rate of increase or decrease is constant, the function will be a straight line. If not, it will be curved. If the curve of the line becomes steeper over time, then the function is changing at an increasing rate; if it becomes flatter, then the function is changing at a decreasing rate.
If you don't see how these patterns make sense, it really helps to test out a bunch of different functions to see how they translate into graphs. Plot out the points and notice patterns in the way they look. Then use those patterns to generalize about the behavior of functions in certain situations.
Once you know how the graphs of functions behave in general, you can convert sentences like 'the function increases at a constant rate in the first quadrant' into graphs. The key to success is letting go of exact numbers and just focusing on behavior.
To show you how it's done, here's an example question:
A function decreases at a constant rate throughout the second quadrant. When it reaches the origin, it begins to increase at an increasing rate. Sketch one potential graph of the function.
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To make this more manageable, we'll break this down into two chunks. First, we'll deal with the part before the origin. Then, we'll tackle the part after the origin.
The problem says that the function decreases at a constant rate throughout the second quadrant. 'Decreases' means that the function will slope downward from left to right, and decreasing at a constant rate means that it's a straight line. So we'll draw a downward-sloping straight line in the second quadrant, stopping at the origin.
Now look at the second part of the problem. When it reaches the origin, it begins to increase at an increasing rate. 'Increasing' means that the line is sloping up from left to right, and we're increasing at an increasing rate, so the line will be getting steeper as it moves to the right.
Completed Example Graph
Tah-dah! That wasn't so bad, was it? It's just putting together the pieces you already learned above.
In this lesson, you learned about sketching graphs of functions based on a verbal description of the function's qualitative features.
Qualitative features are a function's general patterns of behavior, not the specific points on the graph. When you draw a graph based on the qualitative features, you'll be estimating - that's OK.
Just remember how to translate words into graphs:
Increasing functions slant up from left to right, while decreasing functions slant down.
The steeper a line, the greater the rate of increase or decrease.
Curved lines show an increasing or decreasing rate of change. In general, if the line gets steeper, the rate of change is increasing, and if the line gets flatter, the rate of change is decreasing.
Now test yourself on the quiz questions to see how well you've got it!
After reviewing this lesson, you'll have the ability to:
Define qualitative features of a function
Explain how to translate words into graphs based on the function's qualitative features
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