In Exercises 81–82, graph each linear function. 6x-5f(x) - 20 = 0

Verified step by step guidance

1. Rewrite the given equation in the form of a linear function: \(6x - 5f(x) - 20 = 0\).
2. Isolate \(f(x)\) by adding \(5f(x)\) to both sides: \(6x - 20 = 5f(x)\).
3. Divide every term by 5 to solve for \(f(x)\): \(f(x) = \frac{6}{5}x - 4\).
4. Identify the slope and y-intercept from the equation \(f(x) = \frac{6}{5}x - 4\). The slope is \(\frac{6}{5}\) and the y-intercept is \(-4\).
5. Plot the y-intercept \((0, -4)\) on the graph, then use the slope \(\frac{6}{5}\) to find another point by rising 6 units and running 5 units from the y-intercept. Draw the line through these points.

Key Concepts

Linear Functions

A linear function is a mathematical expression that creates a straight line when graphed. It can be represented in the form f(x) = mx + b, where m is the slope and b is the y-intercept. Understanding linear functions is essential for graphing, as it allows one to identify how changes in x affect the value of f(x).

Graphing Linear Equations

Graphing linear equations involves plotting points that satisfy the equation on a coordinate plane. The equation provided can be rearranged into slope-intercept form to easily identify the slope and y-intercept, which are crucial for drawing the line accurately. This process helps visualize the relationship between the variables.

Solving for f(x)

To graph the function given in the equation 6x - 5f(x) - 20 = 0, one must first isolate f(x). This involves rearranging the equation to express f(x) in terms of x. Understanding how to manipulate equations is vital for determining the function's values and subsequently plotting them on a graph.