Problem 82 - College Algebra Essentials

Question

will help you prepare for the material covered in the next section. In each exercise, graph the linear function. $$ f(x)=-\frac{2}{3} x $$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = -\frac{2}{3}x\) is a straight line passing through the origin and with a slope of -2/3.

Step 1: Identify important components

The given function is \(f(x) = -\frac{2}{3}x\). Here, there is no y-intercept and the slope of the function is -2/3. The equation could also be written in the standard format of \(y = mx + c\) where \(m\) is slope (in our case -2/3) and \(c\) is y-intercept (in our case 0).

Step 2: Plot the y-intercept

Now start from the y-intercept on the graph, which is 0. To denote this, make a small dot on the origin 'O' of the graph.

Step 3: Plot another point using the slope

The slope -2/3 means that for every 3 units increase in x, y decreases by 2 units. Thus, from the y-intercept, we can go right 3 units (as x is increasing) and down 2 units (as y is decreasing) to determine a new point on the line.

Step 4: Draw the line

Finally, draw a straight line through the two points. The graph of the function \(f(x) = -\frac{2}{3}x\) is this line.

Key Concepts

Understanding the Slope

The slope of a line is a crucial concept in graphing linear functions. It indicates the direction and steepness of the line. In the function you are working with, the slope is \(-\frac{2}{3}\). This means that for every 3 units you move horizontally to the right, the line will drop vertically by 2 units.

The slope can determine whether a line is increasing, decreasing, or constant:

• Positive slope: line rises as you move from left to right.
• Negative slope: line falls as you move from left to right. This is the case for \(-\frac{2}{3}\).
• Zero slope: line is perfectly horizontal.
• Undefined slope: line is vertical.

Understanding the slope will help you plot additional points and ensure your line is graphically accurate.

Identifying the Y-Intercept

The y-intercept is the point where the line crosses the y-axis of the coordinate plane. For the function \(f(x) = -\frac{2}{3}x\), the y-intercept is 0. This is because there is no constant term added to the expression.

You can think of the y-intercept as the starting point when graphing a function. This is typically where you begin plotting your line on the graph. For instance, at the y-intercept of 0, you place a dot at the origin point (0,0).

The y-intercept is an important concept as it provides a fixed point that helps in accurately drawing the line alongside the slope.

Exploring the Coordinate Plane

The coordinate plane is a two-dimensional surface where we plot points and graph lines. It consists of two axes: the horizontal x-axis and the vertical y-axis. Each point on this plane is represented as a pair of numbers \((x, y)\).

To graph a linear function, we utilize both axes to determine the placement of points:

• The x-axis is where we measure horizontal movement.
• The y-axis measures vertical movement.

When graphing the function \(-\frac{2}{3}x\), you begin at the y-intercept on the y-axis and use the slope to plot your line. The coordinate plane provides a visual way to interpret algebraic equations and understand the behavior of linear functions.

Simplifying Algebraic Equations

Algebraic equations are the expressions that define relationships between variables. In the context of linear functions, these relationships form straight lines when graphed. The function given, \(f(x) = -\frac{2}{3}x\), exemplifies a linear equation.

To understand it fully, break it down to the form \(y = mx + c\). Here, \(m\) denotes the slope, and \(c\) the y-intercept. In your function, the slope \(m\) is \(-\frac{2}{3}\) and the y-intercept \(c\) is 0. This makes it simple to graph as you start at the y-intercept and use \(m\) to find other points.

Algebraic equations allow us to represent complex relationships in a clear and concise format, and being able to rearrange and interpret them is vital in solving and graphing linear functions.