Recognizing Linear Functions - Pre Algebra

1. Is the function in this table linear or nonlinear? $x$: 1, 2, 3, 4; $y$: 3, 7, 11, 15.

   Linear. Each time $x$ increases by 1, $y$ increases by 4.

2. Is the function in this table linear or nonlinear? $x$: 1, 2, 4, 5; $y$: 3, 5, 9, 11.

   Linear. The rates $\frac{2}{1}$, $\frac{4}{2}$ and $\frac{2}{1}$ all equal 2, even though the $x$-steps are uneven.

3. Which equation is linear: $y = \frac{4}{x}$ or $y = \frac{x}{4}$?

   $y = \frac{x}{4}$, which equals $\frac{1}{4}x$. In $y = \frac{4}{x}$ the variable is in the denominator.

4. What is the rate of change for this table? $x$: 0, 2, 4, 6; $y$: 5, 8, 11, 14.

   $\frac{3}{2}$. The $y$-values rise by 3 each time $x$ rises by 2, so the function is linear.

5. What is the rate of change of the function $y = 7$?

   0. The output never changes, so its graph is a horizontal line.

6. Which equation is not linear: $y = 2x$, $y = 5 - x$ or $y = x^3$?

   $y = x^3$. A linear function has no power of $x$ other than 1.

7. What shape is the graph of a linear function?

   A straight line.

8. Which equation is linear: $y = 3x - 2$ or $y = x^2 + 1$?

   $y = 3x - 2$. The $x^2$ term makes $y = x^2 + 1$ nonlinear.

9. Is the function in this table linear or nonlinear? $x$: 1, 2, 3, 4; $y$: 2, 4, 8, 16.

   Nonlinear. The changes in $y$ are 2, 4 and 8, which are not constant; $y$ doubles each time.

10. Is the function in this table linear or nonlinear? $x$: 1, 2, 3, 4; $y$: 1, 4, 9, 16.

    Nonlinear. The changes in $y$ are 3, 5 and 7, which are not constant.

11. What feature of a table of values shows that a function is linear?

    A constant rate of change: equal changes in $x$ always give equal changes in $y$.

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