Michigan Grade 8 Math

Do two matching outputs make the rules the same?

Two mathematical rules assign one output to every input x. Rule A is y = 3x + 2. Rule B is y = x squared + 2. Compare their outputs for x equal to 0, 1, 2, and 3. Explain which rule is linear and whether matching at some inputs proves the rules are identical. These are abstract functions, not measurements from a real device.

Explain the solution

For Rule A, the four outputs are 2, 5, 8, and 11. Equal input increases of one produce equal output increases of three. Its equation has the linear form y = mx + b, with m equal to three and b equal to two. Its graph is a straight line. For Rule B, the outputs are 2, 3, 6, and 11. The successive increases are one, three, and five, so the equal input steps do not have a constant output change. The points (0, 2), (1, 3), and (2, 6) are not on one straight line. Rule B is a function because every input has one output, but it is not linear.

Notice the reasoning

Both rules give two at x = 0 and eleven at x = 3. Those agreements do not prove equivalence: at x = 1, their outputs differ. A learner may call any increasing pattern linear. Ask them to compare equal input changes, not only whether the outputs grow. A finite table that appears linear also does not establish every unspecified output; here the given equations define the rules. Rule A’s equation supplies the relationship beyond the four displayed inputs. The squared term in Rule B produces a nonlinear relationship, and the displayed points provide a concrete counterexample to a straight-line claim.

Try a related task

Use x = 4 to check the rules again. Rule A gives fourteen, while Rule B gives eighteen. Plot the displayed ordered pairs with equally spaced axes and label each rule. Extend the straight line for Rule A according to its equation. For Rule B, remember that connecting a few points with straight segments is only a sketch and does not turn the squared rule into a linear function. Ask the student to create another linear rule with an initial value of two and a different constant change, then explain which parts of its table and graph would change.

Selected references from Michigan mathematics standards

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