Question

a line is drawn through (-4, 3) and (4, 3). which describes whether or not the line represents a direct variation?

• the line represents a direct variation because $-\frac{4}{3} = \frac{4}{3}$.
• the line represents a direct variation because it is horizontal.
• the line does not represent a direct variation because it does not go through the origin.
• the line does not represent a direct variation because $-4(3) \

eq 4(3)$.

Explanation:

Step1: Recall direct variation definition

A direct variation relationship is of the form \( y = kx \), where \( k \) is a constant. The graph of a direct variation must pass through the origin \((0,0)\) because when \( x = 0 \), \( y=0\).

Step2: Analyze the given line

The line passes through \((-4, 3)\) and \((4, 3)\). Let's find the equation of this line. The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{3 - 3}{4 - (-4)}=\frac{0}{8} = 0\). So the equation of the line is \( y=3\) (since it's a horizontal line with \( y\)-value 3 for all \( x\)).

Step3: Check if it passes through origin

For the origin \((0,0)\), when \( x = 0\), \( y=3
eq0\). So the line \( y = 3\) does not pass through the origin. Since direct variation graphs must pass through the origin, this line does not represent a direct variation.

Now let's analyze each option:

• Option 1: \(-\frac{4}{3}=\frac{4}{3}\) is false, and this reasoning is incorrect.
• Option 2: A horizontal line (unless it's \( y = 0\)) does not represent direct variation. Direct variation with \( k = 0\) would be \( y=0\), but here \( y = 3\), so this is wrong.
• Option 3: Correct, because direct variation requires passing through the origin, and this line does not.
• Option 4: The reasoning \( - 4(3)

eq4(3)\) is irrelevant to direct variation definition.

Answer:

The line does not represent a direct variation because it does not go through the origin. (Third option: The line does not represent a direct variation because it does not go through the origin.)