Question

based on the table list if the function is an example of direct variation.
x:4,8,12,16
y:-1,-2,-3,-4
If the function is a direct variation name the constant and if not name the slope
write the equation in slope intercept form

Answer 1

To determine if the function is an example of direct variation, we need to check if the ratio of \( y \) to \( x \) is constant. In a direct variation, the relationship can be expressed as \( y = kx \), where \( k \) is a constant.

We can calculate \( \frac{y}{x} \) for the given pairs:

1. For \( x = 4 \), \( y = -1 \):
\( \frac{y}{x} = \frac{-1}{4} = -\frac{1}{4} \)

2. For \( x = 8 \), \( y = -2 \):
\( \frac{y}{x} = \frac{-2}{8} = -\frac{1}{4} \)

3. For \( x = 12 \), \( y = -3 \):
\( \frac{y}{x} = \frac{-3}{12} = -\frac{1}{4} \)

4. For \( x = 16 \), \( y = -4 \):
\( \frac{y}{x} = \frac{-4}{16} = -\frac{1}{4} \)

Since the ratio \( \frac{y}{x} \) is constant and equal to \(-\frac{1}{4}\), this is an example of direct variation.

### Constant:
The constant \( k \) for this direct variation is \( -\frac{1}{4} \).

### Equation:
The equation of the direct variation can then be written as:

\[
y = -\frac{1}{4}x
\]

Now we will write it in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Since there is no constant added (i.e., the intercept is 0), the slope-intercept form remains:

\[
y = -\frac{1}{4}x + 0
\]

So the equation in slope-intercept form is:

\[
y = -\frac{1}{4}x
\]