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Common Core Standards: Math

Grade 8

Functions 8.F.B.5

5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

There's an old story about three blind men who encounter an elephant for the first time and try to describe it. The first finds himself in contact with the elephant's tail, and says with certainty, "An elephant is just like a rope!" The second comes into contact with the elephant's side, and says with equal certainty, "An elephant is just like a wall!" The third comes into contact with the elephant's trunk, and claims, "An elephant is just like a snake!"

Of course, the story ends there. We never find out what happens when the elephant gets tired of being poked and prodded by these three guys. We're guessing it wasn't pretty. Anyway, the moral of the story has to do with understanding your limited perspective. What each of us sees (or feels, as the case may be) isn't the whole picture.

As useful as that may be, it doesn't really help us out with math. So we're going to offer up a slightly different (but not inaccurate) moral to the story. Sometimes—and, in fact, most of the time—one or two characteristics don't do a great job of describing anything well.

Graphs are the same way. There are many ways students can describe the same graph. Using their knowledge of functions, equations, and graphs and they should be able to describe a graph qualitatively and be able to draw a graph if given a qualitative description. When they come across a graph, they should consider a few specific things.

  1. Is the graph a function? (In other words, does it pass the vertical line test?)
  2. Does the graph increase or decrease? Both? Neither? For which intervals? (An increasing graph goes up from left to right, while a decreasing graph goes down.)
  3. Is the graph continuous? (Can they trace the entire graph without lifting their pencil from the paper?) Is it linear, quadratic, absolute value, or something else?
  4. Does it have any turning points? (Those are exactly what they sound like: points where the graph changes direction—like the bottom point of a U-shaped parabola.)

These are just some of the many characteristics that can be used to describe what a graph looks like. And, like those blind guys and the elephant, each gives just a hint as to the entire description of the graph.

Drills

  1. Is the graph of the equation x2 + y2 = 9 a function?

    Correct Answer:

    No, it fails the vertical line test

    Answer Explanation:

    It should be obvious that the graph is neither absolute value nor linear. If you try to come up with ordered pairs, you can get two with the same x value: (0, 3) and (0, -3). That means the graph would fail the vertical line test. It's not a quadratic or a function at all. It's actually the equation for a circle.


  2. Does the graph of the equation y = 4 – x increase or decrease?

    Correct Answer:

    The graph decreases because the y values decrease from left to right

    Answer Explanation:

    The easiest way to be completely sure is to plug in a few x values and see how the y values change in comparison. Another way is to recognize this is the same as y = -x + 4, an equation for a line with a slope of -1, which means the y values will decrease as the x values increase. In other words, (D) is correct because (B) and (C) are wrong for this function and although the x values increase from left to right, that's just how the coordinate plane works. It has nothing to do with the graph.


  3. Is the graph of y = x a function?

    Correct Answer:

    Yes, because it passes the vertical line test

    Answer Explanation:

    If you graph it, you get a line that gradually increases from left to right. If you were to draw a vertical line anywhere, it would only hit the graph once. That is what makes it a function: there's only one y value for every x. The other answer choices are wrong because the horizontal line test is not used to test whether graphs are functions, not all relations that can be graphed are functions, and it is possible for a function to decrease.


  4. Which of the following functions is constantly increasing?

    Correct Answer:

    y = x – 7

    Answer Explanation:

    While it's always useful to plot points and see how the y values change for certain x inputs, it might not give the big picture. We can see that (A) is quadratic, (B) and (C) are linear, and (D) is an absolute value function. Quadratic and absolute value functions form U's and V's, which means they decrease and increase. Only the linear function with the positive slope will constantly increase.


  5. Which of the following functions is constantly decreasing?

    Correct Answer:

    x + y = 3

    Answer Explanation:

    While some might be inclined to go for (D) because its y values will only be negative, quadratics increase and decrease regardless of how many negative signs we put in front of them. It should be fairly obvious that (A) increases (since it's a line with a positive slope), but (C) has a negative slope when written in standard form. As for (B), it's a positive cubic function so it'll increase, decrease, and then increase again or just increase. Regardless, the only constantly decreasing function is (C).


  6. Which of the following has no turning points?

    Correct Answer:

    Linear

    Answer Explanation:

    This might be obvious by the words we use to describe turning points and linear functions. Lines and turning just don't go together. Since quadratic functions make parabolas (that look like U's or upside-down U's), they have turning points. Absolute value functions also have turning points (since they look like V's). Only lines have no turning points. They keep going in the same direction…forever.


  7. Which of the following is a continuous function?

    Correct Answer:

    Answer Explanation:

    Here, we need to be sure our graph fulfills two criteria: it has to be a function and it has to be continuous. We can apply the vertical line test to make sure that the graphs we're looking at are functions. Unfortunately, only (A) and (C) make the cut. Continuous functions mean they're defined for all values, so there should be no gaps or breaks in the curve. Since (A) has a break, the only continuous function is (C).


  8. Which of the following functions has a maximum?

    Correct Answer:

    Answer Explanation:

    Since vertical lines keep going up and down forever, we know they don't have a maximum or minimum. Positive absolute value functions look like giant V's, which means they have minimum values, not maximum values. Horizontal lines have no minimums or maximums either since they're just one y value. Only when the x2 term is negative will a parabola have a maximum value, so (C) is right.


  9. Which of the following sets of points does not form a function?

    Correct Answer:

    (0, 0), (0, 1), (0, 2)

    Answer Explanation:

    Since a function is a relation in which every x has only one y, the same x value must result in the same y value. All of the x values in (B), (C), and (D) are different and produce different y values. Having the same x value and different y values in (A), however, means that those points cannot form a function.


  10. Which of the following is a linear increasing function?

    Correct Answer:

    Answer Explanation:

    All the graphs here are functions since they pass the vertical line test. We can narrow the graphs down because only (B) and (C) are linear. The graph in (A) increases and decreases and is very clearly not linear. The graph in (D) is constantly increasing, but not in a linear fashion. Of the two lines, (C) is increasing and (B) is horizontal and therefore constant. Even though the graph in (C) is below the x-axis and negative, it is increasing because as we move further and further to the right, the y values get more and more positive.


More standards from Grade 8 - Functions