Question

which graph represents the function ( f(x) = \frac{1}{4}x - 6 )?
options: a, b, c (with corresponding graphs)

Explanation:

To determine the graph of \( f(x)=\frac{1}{4}x - 6 \), we analyze the linear function's properties:

Step 1: Identify the slope and y - intercept

The equation is in slope - intercept form \( y = mx + b \), where \( m=\frac{1}{4} \) (slope) and \( b=-6 \) (y - intercept).

• A positive slope (\( m=\frac{1}{4}>0 \)) means the line rises from left to right.
• The y - intercept \( b = - 6 \) means the line crosses the y - axis at \( (0,-6) \).

Step 2: Analyze the options

• Option A: The line has a positive slope (rises left - to - right) and intersects the y - axis at a negative value (consistent with \( b=-6 \)) and the x - axis at a positive value (we can find the x - intercept by setting \( y = 0 \): \( 0=\frac{1}{4}x-6\Rightarrow\frac{1}{4}x = 6\Rightarrow x = 24 \), but visually, the positive slope and negative y - intercept match the form of \( y=\frac{1}{4}x - 6 \)).
• Option B: The line has a negative slope (falls left - to - right), which contradicts \( m=\frac{1}{4}>0 \).
• Option C: The line has a negative slope (falls left - to - right), which contradicts \( m=\frac{1}{4}>0 \).

Answer:

A (the graph with positive slope, crossing the y - axis at a negative value)