Question

Determine whether each table or equation represents an inverse or a direct variation. Explain.

$x$	1	4	8	12
$y$	2	8	16	24

Short answer

The table represents a direct variation.

Step-by-step solution

Step 1. State the concept of direct variation and inverse variation.

The relationship of the form $y=kx$is called direct variation.

Relationship of the form $xy=k$or $y=\frac{k}{x}$where $x,y\ne 0$for some nonzero constant $k$is known as inverse variation i.e., $y$varies inversely as $x$.

Step 2. Check for inverse variation. 

Find $xy$each ordered pair from the table and get,

$$\begin{gathered} 1\cdot 2=2 \\ 4\cdot 8=32 \\ 8\cdot 16=128 \\ 12\cdot 24=288 \end{gathered}$$

The product is not constant so the table doesn’t represent an inverse variation.

Step 3. Check for direct variation.

Find $\frac{y}{x}$each ordered pair from the table and get,

$$\begin{gathered} \frac{2}{1}=2 \\ \frac{8}{4}=2 \\ \frac{16}{8}=2 \\ \frac{24}{12}=2 \end{gathered}$$

Since, the $\frac{y}{x}=2$, for all given values of $x$and $y$, so the table represents the direct variation of $y=2x$.