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Common Core Standards: Math

Grade 8

Functions 8.F.A.3

3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1, 1), (2, 4) and (3, 9), which are not on a straight line.

The artist Paul Klee once said, "A line is a dot that went for a walk." You know what? He was right. A line really is just a collection of points. When a function forms a line, that function is called linear.

Students should already be more or less familiar with linear equations. At least to the point where they get a slight déjà-vu when they see the equation y = mx + b. All you need to do now (aside from refresh their memory) is tell them that this equation for a line is the same as a linear function. As long as x stays with a power of 1, the function will be linear.

Students should be able to recognize linear functions in standard form (like y = 4x + 5 and -½x – 7) and graph them as straight lines. Some points that go on walks keep going on and on in a straight line forever. Other points prefer to travel along the winding scenic route.

 

Functions that aren't lines are grouped into the nonlinear category. Students should be able to recognize functions that aren't linear. For instance, the equation y = x2 is a function because each x value produces one y value. Yet, it doesn't fit the pattern of y = mx + b. If we graph the function, we end up with a curve called a parabola, shaped like a giant U.

Students should recognize quadratics and absolute value functions as nonlinear. (The graphs of both look like points started going one way and then changed their minds.) While they are still functions (assure students that the definition of "function" hasn't changed!), they are most definitely not lines. And that should be clear from both the graphs and the equations.

If students aren't sure, they can tell whether a function is linear by looking at the equation and seeing whether it fits into y = mx + b. Even an equation like 3x + 9y = 0 can be rearranged to fit the standard form, but y = x2 + 2x + 1 never will. If they still need more convincing, they can plot the points and see whether or not the graph is a straight line.

Whether linear or nonlinear, students should know that equations and graphs aren't always functions. The easiest way to determine whether or not a graph is that of a function is the vertical line test. If a vertical line can be drawn anywhere on the graph without hitting the line or curve more than once, the graph is a function. If not, what we have is not a function, but rather a point with no sense of direction.

Drills

  1. How can we be sure that the function y = 4x + 2 is linear?

    Correct Answer:

    Both (A) and (B)

    Answer Explanation:

    The equation y = 4x + 2 fits into the standard form of a linear equation, y = mx + b. All we need to do is substitute m = 4 and b = 2 and the equations are the same. Of course, if a function is linear, it should logically form a line when we graph it on the coordinate plane. Lucky for us, that's exactly what happens. While the slope (4) is two times the y-intercept (2), that's not necessary for a function to be linear. Both (A) and (B) are the right reasons, which means (D) is the answer.


  2. Which of the following points does not satisfy the function y = 3x + 2?

    Correct Answer:

    (-1, -1)

    Answer Explanation:

    To test the points, plug the first number in for x and see if y turns out to be the second number. The only point this is untrue for is (D) because y = 3(-5) + 2 = y = -15 + 2 = -13. If (D) appears to be correct, the 2 at the end may have been accidentally subtracted instead of added.


  3. Which of the following equations is a line, but not a function?

    Correct Answer:

    x = -10

    Answer Explanation:

    We know that if the equation fits the form y = mx + b, it's a line. Since (A) does not fit this form, it is not a line. The remaining three options are lines, but only (C) and (D) are functions. This is because (B) has a constant x (input) value but no specified y (output) value. For one input (x = -10), there will be an infinite number of outputs. This results in a vertical line that is not a function.


  4. Which of the following sets of points forms a linear function?

    Correct Answer:

    (1, 2), (3, 6), (4, 8)

    Answer Explanation:

    In a line, the rate of change between any two points should be constant. This isn't the case for (B), (C), or (D). Each of the three points in (A), however, fit into the equation y = 2x. If this isn't clear from the numbers, plot the points and see if it's possible to fit a single line through all three of them. It should only be possible for (A).


  5. Which of the following equations is not a function?

    Correct Answer:

    x = y2

    Answer Explanation:

    All of these equations may look a bit foreign at first glance. If we go one by one, we can see that (A) is nonlinear, but still a function. Even with its power of 3 and absolute value bars, (B) is also a nonlinear function. While (C) and (D) look strange because the equations are in terms of x rather than y, we can change (D) into y = mx + b form by subtracting 1 and dividing by 2. Graphing (C), however, will look like a sideways parabola. If we have an input of 1, y could be either 1 or -1. Two outputs for one input is a huge red flag!


  6. Which of the following would be result in the graph of a linear function?

    Correct Answer:

    The set of points where the y was one-fifth the x value

    Answer Explanation:

    We can interpret each of the choices here as graphs or equations of some sort. Option (A) would take the shape of a parabola since the ball would travel in an arc—not a line—through the air. The words "absolute value" typically don't denote a linear function and in this case, the equation |y| = x isn't even a function (because x = 3 can result in y = 3 or -3). Graphing (C) will form a circle with a radius 5, which is definitely not a function. The only option left is (D), which results in the equation y = ⅕x, a function that fits the standard form of a line.


  7. Which set of points is on the linear function y = -x + 4?

    Correct Answer:

    (0, 4), (1, 3), (2, 2)

    Answer Explanation:

    To see whether or not the points match the equation, we can plug in the x values and see if the correct y values pop out. While it's true that y = -(0) + 4 = 4, the only value of y that results for x = 1 is y = -(1) + 4 = 3, which means (D) is the right set of points. We can double-check this too by plugging in 2 for x and verifying that y = -(2) + 4 = 2.


  8. Which of the following is a nonlinear function?

    Correct Answer:

    y = ¼x4

    Answer Explanation:

    We're looking for an equation that is still a function, but doesn't fit into the standard y = mx + b form of a line. While none of these equations look like that, it only takes a little rearranging to see that (A) is linear. The absolute value bars in (D) are only around the -13, so the equation is really just y = 13x, which is linear too. Tricky. Of the remaining two choices, neither is linear, but only (C) is a function.


  9. Which of the following is true about the equation x = 17y + 17?

    Correct Answer:

    It passes the vertical line test

    Answer Explanation:

    Before we can look at equation x = 17y + 17, we might want to try rearranging it into standard form, if possible. Subtracting 17 and then dividing by 17 gives us the equation y = 117x – 1. This is a linear function in standard form, where the slope is 117 and the y-intercept is -1. This tells us that (A) and (C) are both wrong. Vertical lines aren't functions, so (D) is incorrect as well. Passing the vertical line test means the equation is a function, so (B) is true.


  10. Which of the following is true about the function y = 3x2?

    Correct Answer:

    It has a rate of change that constantly changes

    Answer Explanation:

    It should be clear from the get-go that the equation y = 3x2 is not linear, thanks to the exponent that's staring us in the face. It's a nonlinear function—but still a function. Mathematically, the only way we can get rid of an exponent is a square root, and (A) doesn't have that so the two aren't equivalent. If we plug in a few points, we should be able to tell that it is a function and (B) is wrong because it will pass the vertical line test. Option (D) is almost right, except a sideways parabola isn't a function while y = 3x2 is. That leaves (C), which is true because lines have rates of change that stay constant, but parabolas don't.


More standards from Grade 8 - Functions