math-2017:8.F.3
math-2017:8.F.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.
Learning goal: Interpret the equation y = mx + b as defining a linear function, explain why the graph of every such function is a straight line, and give examples of functions that are not linear, showing with specific points why their graphs are not straight lines.
What y = mx + b says. The equation y = mx + b defines a function: for each input x there is exactly one output, found by multiplying by m and adding b. Such a function is called linear. The number m is the rate of change and the number b is the output when x = 0. For example, y = 3x − 2 has m = 3 and b = −2. Its inputs 0, 1, 2, 3 give the outputs −2, 1, 4, 7, and the ordered pairs are (0,−2), (1,1), (2,4), (3,7).
Why the graph is a straight line. Look at what happens to the output each time the input increases by 1. From x = 0 to x = 1 the output rises from −2 to 1, a change of 3. From x = 1 to x = 2 it rises from 1 to 4, again 3. From x = 2 to x = 3 it rises from 4 to 7, again 3. The change in y per unit change in x is the same, m, everywhere. Geometrically, moving 1 to the right always moves you the same amount up or down, so the points line up along a single straight path. Between any two different points on the graph, the ratio of the change in y to the change in x is m, no matter which two points you choose. A constant rate of change is exactly what makes a graph straight.
The letters m and b can be any numbers, including 0, negative numbers and fractions. If m is negative, the line falls from left to right. If m = 0, the equation becomes y = b, whose graph is a horizontal line; that is still a linear function, since each input gets exactly one output and the graph is straight. If b = 0, the line passes through the origin and, provided m is not 0, the function is also a proportional relationship. An equation may be disguised: 4x + 2y = 10 can be rearranged to y = −2x + 5, so it defines a linear function with m = −2 and b = 5. The test is whether the equation can be written in the form y = mx + b.
A vertical line is not a function. The equation x = 4 has a straight-line graph, but every point on it has input 4, so the input 4 has infinitely many outputs. It is not a function, and it cannot be written as y = mx + b. A straight graph is not enough on its own; the relationship must also assign one output to each input.
Functions that are not linear. Many functions cannot be written as y = mx + b. The clearest example is the area of a square as a function of its side length: A = s². This is a function, because each side length gives exactly one area. But its graph contains the points (1,1), (2,4) and (3,9), and those three points are not on a straight line. Check with rates of change: from (1,1) to (2,4), the area rises 3 for a side increase of 1. From (2,4) to (3,9), the area rises 5 for a side increase of 1. The rate of change is not constant, so no single straight line passes through all three points. If you tried the line through (1,1) and (2,4), which has rate 3, it would predict an area of 7 at s = 3, not 9.
Contrast this with the perimeter of a square, P = 4s. Here the perimeter rises by 4 for every unit of side length: (1,4), (2,8), (3,12). This is y = mx + b with m = 4 and b = 0, and its graph is a straight line. Two functions built from the same square, one linear and one not.
Other examples of functions that are not linear: the volume of a cube, V = s³, with points (1,1), (2,8), (3,27), where the successive changes are 7 and 19; the doubling function y = 2ˣ, with points (0,1), (1,2), (2,4), (3,8), where the successive changes are 1, 2 and 4; and y = 1/x for positive x, with points (1,1), (2,0.5), (4,0.25), where the output falls by 0.5 over the first unit but by only 0.25 over the next two units. The absolute value function y = |x| is also not linear over all inputs: (−1,1), (0,0) and (1,1) do not lie on one line, because the graph turns at the origin, even though each half on its own looks straight.
How to test any function. Make a table with equally spaced inputs. Compute the change in output between consecutive rows. If every change is the same, the points you computed lie on a straight line and are consistent with a linear function; to be sure the whole function is linear you need its rule in the form y = mx + b, since a table shows only some of its points. If the changes differ, the function is definitely not linear, and naming two of the differing changes is a complete justification: one pair of unequal rates rules a straight line out, while any number of equal rates cannot rule a curve in. Alternatively, compute the rate of change between two different pairs of points; if the two rates differ, the function is not linear.
A fully worked model. Is y = x(x + 1) a linear function? Compute outputs for x = 1, 2, 3, 4: 1 × 2 = 2, 2 × 3 = 6, 3 × 4 = 12, 4 × 5 = 20. The points are (1,2), (2,6), (3,12), (4,20). Successive changes in y for unit changes in x are 4, 6 and 8. They are not constant, so the points are not on a straight line and the function is not linear. Algebraically, x(x + 1) = x² + x, which contains an x² term and so cannot be written as mx + b. Both the numerical check and the algebraic form agree.
A caution about small tables. Three points that happen to be collinear do not prove a function is linear for every input. The safe claim is that the equation has the form y = mx + b; then the constant rate of change follows for all inputs. When you only have a few points, say "these points are consistent with a linear function" unless you know the rule.
Worked linear check: For y = 3x − 2, the inputs 0, 1, 2, 3 give outputs −2, 1, 4, 7. The consecutive changes are 3, 3 and 3, all equal to m. The rate between (0,−2) and (3,7) is (7 − (−2)) ÷ 3 = 9 ÷ 3 = 3, the same as between (1,1) and (2,4), which is 3 ÷ 1 = 3. Every pair of points gives the same rate, so the graph is a straight line with m = 3 and b = −2.
Worked square area: For A = s², the side lengths 1, 2, 3 give areas 1, 4, 9, so the graph contains (1,1), (2,4) and (3,9). The rate from (1,1) to (2,4) is 3 per unit; from (2,4) to (3,9) it is 5 per unit. Different rates mean the three points are not on one line. The line through (1,1) and (3,9) has rate (9 − 1) ÷ (3 − 1) = 4 and would pass through (2,5), but the actual point is (2,4), which is below that line. The function is not linear.
Worked rearrangement: Does 6x + 3y = 12 define a linear function? Subtract 6x from both sides: 3y = −6x + 12. Divide by 3: y = −2x + 4. This is y = mx + b with m = −2 and b = 4, so it is a linear function whose graph falls 2 units for each unit to the right and crosses the vertical axis at 4. Check with a point: x = 1 gives y = 2, and 6 × 1 + 3 × 2 = 12 as required.
Representation task: On one pair of axes from s = 0 to s = 4, plot the perimeter function P = 4s and the area function A = s² for s = 0, 1, 2, 3, 4. Draw a straight line through the perimeter points with a ruler and check that all five lie on it. Try to do the same for the area points and record why it fails. Under the graph, make a table of the consecutive changes in P and in A and circle the column that is constant.
Independent performance: Write four functions: two that are linear and two that are not. At least one linear function must have a negative m and one must be given in a disguised form such as 2x + y = 7. At least one non-linear function must come from geometry (an area, a volume, a number of diagonals) and one from a rule you invent. For each, make a table for four equally spaced inputs, compute the consecutive changes in output, and write one sentence that either gives m and b or names two different rates of change as the reason the function is not linear. Success criteria: every output is correct; disguised linear equations are rearranged correctly; non-linear claims cite specific points; no function is called linear on the basis of its graph looking straight without the rate check.
An adult may read the prompts aloud. Answer independently with words, objects, drawings or a demonstration before opening the answer guide. Award 1 point for a correct result or claim and 1 point for a matching explanation or demonstration; where a question asks only for a result (a corrected sentence, a spelling, a count), a fully correct result earns both points, and the answer guide shows what a complete response contains. Record prompting separately. The total (two points per question) is local practice feedback, not an Ohio proficiency level.
Is y = −3x + 7 a linear function? If so, give m and b and three points on its graph.
Yes. It is in the form y = mx + b with m = −3 and b = 7. The points (0,7), (1,4) and (2,1) are on the graph (any three points that satisfy the equation, such as (3,−2), are equally correct), and the output falls by 3 for each unit increase in x, so the graph is a straight line that slopes downward.
Give an example of your own of a function that is not linear, and use three of its points to show that it is not linear.
Answers vary. For example, y = x² + 1 gives (0,1), (1,2) and (2,5). From x = 0 to 1 the output rises 1; from 1 to 2 it rises 3. A linear function has the same rate of change between every pair of points, so these three points are not on one straight line and the function is not linear. Any function with a correct table and two unequal rates of change over equal steps of x (or a rule that cannot be written as y = mx + b, shown with points) is acceptable.
Is y = 5 a linear function? Explain.
Yes. It can be written as y = 0x + 5, so m = 0 and b = 5. Each input has exactly one output, 5, and the graph is a horizontal straight line through (0,5). A rate of change of 0 is still a constant rate.
Compute y = (x + 1)² for x = 0, 1, 2 and 3 and decide whether the function is linear.
y = 1, 4, 9 and 16. The changes for unit increases in x are 3, 5 and 7. Since the rate of change is not constant, the points (0,1), (1,4), (2,9) and (3,16) are not on a straight line and the function is not linear; algebraically, (x + 1)² = x² + 2x + 1 cannot be written as mx + b.
Is the tripling function y = 3ˣ linear? Use the inputs 0, 1, 2 and 3.
No. The outputs are 1, 3, 9 and 27, so the consecutive changes are 2, 6 and 18. A linear function would change by the same amount for each unit increase in x. The changes grow, so the graph curves upward and the function is not linear.
Does 9x + 3y = 6 define a linear function? Rewrite it if so.
Yes. Subtracting 9x and dividing by 3 gives y = −3x + 2, which has the form y = mx + b with m = −3 and b = 2. The graph is a straight line crossing the vertical axis at 2 and falling 3 for each unit to the right.
The graph of x = −3 is a straight line. Is x = −3 a linear function? Explain.
No. Every point on that vertical line has input −3, so the input −3 has infinitely many outputs. A function must assign exactly one output to each input, and the equation cannot be written as y = mx + b. A straight graph is not enough; the relationship must also be a function.
Tickets cost 6 dollars each, so the cost of n tickets is C = 6n. Is this a linear function, and what are m and b?
Yes. C = 6n is y = mx + b with m = 6 and b = 0. The cost rises by 6 dollars for every ticket, so the points (1,6), (2,12), (3,18) lie on a straight line through the origin (only whole numbers of tickets make sense, so the graph is those points on that line).