Question

the graph represents the linear function, (f(x)).

which function is best represented by the graph?

a. (f(x) = \frac{1}{8}x - 3)
b. (f(x) = \frac{1}{8}x + 3)
c. (f(x) = -\frac{1}{8}x - 3)
d. (f(x) = -8x + 3)

Explanation:

🆕 New Concept Discovered: Slope-Intercept Form of a Linear Equation
How a line's slope and y-intercept define its graph

Step 1: Identify the y-intercept

The \(y\)-intercept is the point where the graph crosses the vertical \(y\)-axis. Looking at the graph, the line crosses the vertical axis at \( (0, -3) \).

This means the \(y\)-intercept, represented by \(b\) in the slope-intercept form \(f(x) = mx + b\), is:

$$ b = -3 $$

Step 2: Determine the slope

The slope \(m\) measures the steepness and direction of the line.

1. Direction: Since the line goes down from left to right, the slope must be negative (\(m < 0\)).
2. Rise over Run: Find another clear point on the grid that the line passes through. The line passes through \( (0, -3) \) and \( (8, -4) \).

Using the slope formula:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

$$ m = \frac{-4 - (-3)}{8 - 0} = \frac{-1}{8} = -\frac{1}{8} $$

Step 3: Write the function

Substitute the slope \(m = -\frac{1}{8}\) and \(y\)-intercept \(b = -3\) into the slope-intercept form:

$$ f(x) = -\frac{1}{8}x - 3 $$

This matches option C.

Answer:

C \(f(x) = -\frac{1}{8}x - 3\)