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. Identify two points on the line. For example, the points (0, 10) and (5, 0) are on the line.
2. Calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). Substitute the coordinates of the two points into the formula.
3. Identify the y-intercept (b) from the graph, which is the point where the line crosses the y-axis. In this case, it is (0, 10).
4. Identify the x-intercept from the graph, which is the point where the line crosses the x-axis. In this case, it is (5, 0).
5. Write the equation of the line in slope-intercept form, y = mx + b, using the slope and y-intercept identified.

Key Concepts

Slope

The slope of a linear function represents the rate of change of the function, indicating how much the y-value changes for a unit change in the x-value. It is calculated as the rise over run, or the change in y divided by the change in x between two points on the line. A positive slope indicates an upward trend, while a negative slope indicates a downward trend, as seen in the provided graph.

Intercepts

Intercepts are the points where the graph of a function crosses the axes. The y-intercept is the point where the graph intersects the y-axis, indicating the value of the function when x is zero. The x-intercept is where the graph crosses the x-axis, showing the value of x when the function equals zero. Identifying these points is crucial for understanding the behavior of the linear function.

Linear Equation

A linear equation is an algebraic expression that represents a straight line when graphed. It is typically written in the form y = mx + b, where m is the slope and b is the y-intercept. This equation allows us to predict the value of y for any given x, and vice versa. Writing the equation of the line based on the identified slope and intercepts is essential for fully defining the linear function.