Problem 42 - College Algebra

Question

Give the slope and \(y\) -intercept of each line Whose equation is given. Then graph the r function. \(f(x)--3 x+2\)

Step-by-Step Solution

Answer

The slope of the function \(f(x) = -3x + 2\) is -3 and the y-intercept is 2.

Step 1: Identifying the Slope and Y-Intercept

Looking at the given function \(f(x) = -3x + 2\), the coefficient of \(x\), which is -3, is the slope. The constant, which is 2, represents the y-intercept.

Step 2: Graphing the Function

To graph the function \(f(x) = -3x + 2\), start by plotting the y-intercept at point (0,2) on a graph. From this point, move down 3 units (because of the -3 slope) and to the right 1 unit, placing a second point. Draw the line passing through these points, which represents the given function.

Key Concepts

Understanding Slope

The slope of a line describes its steepness and direction. In the function \(f(x) = -3x + 2\), the slope is represented by the coefficient of \(x\), which is -3. A negative slope means the line falls as you move from left to right. The slope of -3 can be understood as a ratio: for every 3 units you move down vertically, you move 1 unit horizontally to the right.

When analyzing slope, keep in mind:

• Positive slopes rise as you move right.
• Negative slopes descend as you move right.
• If the slope is 0, the line is horizontal.
• If the slope is undefined, the line is vertical.

Understanding the slope is essential for predicting how a line behaves just by looking at its equation. This makes it easier to graph the line accurately.

Exploring the Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. From the equation \(f(x) = -3x + 2\), the y-intercept is 2. This indicates that the line intersects the y-axis at the point (0, 2).

To find the y-intercept in any linear equation in the form \(y = mx + b\):

• The term \(b\) is the y-intercept.
• It tells us the starting point of the line on the graph.

Understanding the y-intercept helps in quickly plotting the starting point of the line on the graph, ensuring an accurate depiction of the function.

Graphing Linear Functions

Graphing a linear function involves using both the slope and the y-intercept to plot and draw the line on a coordinate plane. Starting with the equation \(f(x) = -3x + 2\), the process begins by plotting the y-intercept, which is the point (0, 2).

Here's a simple step-by-step method to graph the function:

• Plot the y-intercept: Find the point where \(x = 0\) on the y-axis, which is (0,2).
• Use the slope: From (0,2), move according to the slope. For a slope of -3, move down 3 units, then move 1 unit to the right.
• Mark the second point: Place a point at this new location.
• Draw the line: Connect these points with a straight line extending in both directions.

This line now represents the graph of the function. The use of slope and y-intercept simplifies the process, ensuring that the graph accurately reflects the function's behavior.