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

A direct variation is a relationship 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 \( m \) between these two points is \( \frac{3 - 3}{4 - (-4)} = \frac{0}{8} = 0 \). So the equation of the line is \( y = 3 \) (since it's a horizontal line with \( y \)-coordinate 3 for all \( x \)).

Step3: Check if it Passes Through Origin

The origin is \((0,0)\). For the line \( y = 3 \), when \( x = 0 \), \( y = 3
eq 0 \). So the line does not pass through the origin.

Step4: Evaluate the Options

• Option 1: \( -\frac{4}{3}

eq \frac{4}{3} \), so this is incorrect.

• Option 2: A horizontal line \( y = 3 \) does not represent direct variation (direct variation has \( y = kx \), not a constant \( y \) unless \( k = 0 \) and \( y = 0 \), but here \( y = 3 \)), so this is incorrect.
• Option 3: Direct variation must pass through the origin. This line \( y = 3 \) does not pass through \((0,0)\), so this is correct.
• Option 4: The condition \( -4(3)

eq 4(3) \) is not relevant to direct variation (direct variation is about \( y = kx \), not product of coordinates), so this is incorrect.

Answer:

The line does not represent a direct variation because it does not go through the origin. (Corresponding to the third option: "The line does not represent a direct variation because it does not go through the origin.")