The graph of a linear function f is shown. (a) Identify the slope, y-intercept, and x-intercept. (b) Write an equation that defines f.

Verified step by step guidance

1. Step 1: Identify the y-intercept by finding the point where the line crosses the y-axis. From the graph, the line crosses the y-axis at (0, 10), so the y-intercept is 10.
2. Step 2: Identify the x-intercept by finding the point where the line crosses the x-axis. From the graph, the line crosses the x-axis at (5, 0), so the x-intercept is 5.
3. Step 3: Calculate the slope of the line using the formula \(\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\). Using the points (0, 10) and (5, 0), substitute to get \(\frac{0 - 10}{5 - 0} = \frac{-10}{5}\).
4. Step 4: Simplify the slope calculation to find the slope value. This will give you the slope of the line.
5. Step 5: Write the equation of the line in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Substitute the slope and y-intercept values found in previous steps to write the equation defining the function \(f\).

Key Concepts

Slope of a Linear Function

The slope measures the steepness and direction of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points. A negative slope indicates the line decreases from left to right, as shown in the graph.

Intercepts of a Linear Function

The y-intercept is the point where the line crosses the y-axis (x=0), representing the function's value when the input is zero. The x-intercept is where the line crosses the x-axis (y=0), indicating the input value that makes the function zero.

Equation of a Line in Slope-Intercept Form

The equation of a line can be written as y = mx + b, where m is the slope and b is the y-intercept. Using the slope and y-intercept from the graph, you can write the function's equation to describe the line algebraically.