Problem 42 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The slope of the line is -3 and the \(y\)-intercept is 2.

Step 1: Identify the slope

The slope of a linear function is the coefficient of \(x\) in the equation. So, in this case, the slope \(m\) is -3.

Step 2: Identify the y-intercept

The \(y\)-intercept is the constant term in the equation of the line, which is the point where the line intercepts with the \(y\)-axis. Thus, in this case, the \(y\)-intercept \(b\) is 2.

Step 3: Graph the linear function

To graph a linear function, first plot the \(y\)-intercept on the y-axis. Here, that would be the point (0, 2). Then, use the slope to determine the rise and run for the next point. A slope of -3 means for every 3 units down (negative slope), move 1 unit to the right. Plot this point starting from the \(y\)-intercept and draw a straight line through these points.

Key Concepts

Slope

When you encounter the term **slope** in linear equations, it refers to the measure of steepness or the inclination of a line. Mathematically, it is denoted by the letter \( m \). In the equation of a linear function, typically represented as \( y = mx + b \), the slope is the coefficient that accompanies the \( x \) variable.

The slope is calculated by the formula:

• \( m = \frac{{\text{{rise}}}}{{\text{{run}}}} \)

This ratio indicates how many units the line goes up or down (rise) for every unit it moves from left to right (run).

In our equation \( f(x) = -3x + 2 \), the slope \( m \) is \(-3\). This indicates a line that decreases steeply, moving downward as it progresses from left to right. **Negative slope** means for each unit you move to the right, you move three units downward. Understanding slope is crucial since it describes the direction and angle at which the line tilts across the graph.

Y-Intercept

The **\(y\)-intercept** is another key element in understanding linear equations. It represents the point where the line crosses the \(y\)-axis of the graph. In the equation \( y = mx + b \), the \(y\)-intercept is given by the constant term \( b \).

It's significant because it's often a reference point when beginning to graph the function. For \( f(x) = -3x + 2 \), the \(y\)-intercept \( b \) is \(2\). This tells us that the line crosses the \(y\)-axis at the point \((0, 2)\). In other words, when \(x = 0\), the value of \(y\) is \(2\).

In a graphical representation, no matter the slope of the line, this point is a fixed starting position. It's especially useful in plotting the graph because it provides a specific location to begin drawing the line before using the slope to determine its path.

Graphing Linear Functions

**Graphing linear functions** is a way to visually represent the equation on a coordinate grid. Let's walk through the steps to graph the function \( f(x) = -3x + 2 \).

• **Step 1:** Start by identifying the \(y\)-intercept. For our equation, the \(y\)-intercept is \((0, 2)\). Place a point on the coordinate plane at this location.

• **Step 2:** Use the slope to find the next point. A slope of \(-3\) means you "rise" \(-3\), effectively moving down 3 units, for every 1 unit you "run" to the right. From the \(y\)-intercept, move downward 3 units and right 1 unit to locate the next point.

• **Step 3:** Plot this new point. Now, draw a straight line going through both points. Extend the line in both directions, and you’ve successfully graphed the linear function.

Graphing provides a clear representation of the function's behavior—where it starts, its direction, and its slope across the grid. By looking at a graph, one can easily assess relationships and intersections with other functions.