Problem 86 - Algebra and Trigonometry

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 linear function \(f(x)=-\frac{2}{3}x\) is a straight line passing through the points (-3, 2), (0, 0), and (3, -2).

Step 1: Understanding the linear function

A linear function is a function whose graph is a straight line. The equation of the linear function in this exercise is \(f(x)=-\frac{2}{3}x\). Linear functions can be written in the form \(y=mx+n\), where \(m\) is the slope of the line and \(n\) is the y-intercept. In this case, the slope \(m\) is -\frac{2}{3} and there's no y-intercept.

Step 2: Generating points from the function

To graph the function, need to derive points from the function that can then be plotted. Let's select a few arbitrary x-values, substitute them into the function to find the corresponding y-values (output of function at point x), and form points as (x, f(x)). Let's select x-values as -3, 0, and 3.

Step 3: Deriving y-values (f(x)) for the selected x-values

Substitute x=-3 into the function: \(f(-3)=-\frac{2}{3}*(-3) = 2\). So, the first point is (-3, 2). Substitute x=0 into the function: \(f(0)=-\frac{2}{3}*0 = 0\). So, the second point is (0, 0). Substitute x=3 into the function: \(f(3)=-\frac{2}{3}*3 = -2\). So, the third point is (3, -2).

Step 4: Plotting the points on a graph

Now that the points (-3, 2), (0, 0), and (3, -2) have been derived, they are going to be plotted on a graph. Mark the points on the graph and draw a straight line through them to represent the function \(f(x)=-\frac{2}{3}x\).

Key Concepts

Graphing Linear Equations

Graphing linear equations is the process of taking an equation and drawing its representation on a coordinate grid. This visual representation is often a straight line, showing all possible solutions to the equation in terms of coordinates (x, y). Linear equations come in the form of \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept that indicates where the line crosses the y-axis.
To create the graph:

• Select a set of x-values, which are the independent variable values you choose to calculate y-values.
• Plug these x-values into the equation to find the corresponding y-values.
• Plot these points on the graph.
• Draw a straight line through the plotted points.

This method shows how all the solutions to the linear equation align in a straight path, confirming its 'linear' nature. Linear equations are important because they can model real-world situations, such as calculating a constant rate or slope.

Slope-Intercept Form

The slope-intercept form is a common way of expressing linear equations. It is represented as \( y = mx + b \). This form is particularly useful because it clearly shows the slope \( m \) and y-intercept \( b \) of the line.

• The **slope** \( m \) tells us how much the y-value changes for each unit change in the x-value. In the equation \( y = -\frac{2}{3}x \), the slope is \(-\frac{2}{3}\), meaning that the line drops 2 units downwards for every 3 units it moves to the right.
• The **y-intercept** \( b \) is the value of y where the line crosses the y-axis. If \( b = 0 \), like in the example \( y = -\frac{2}{3}x \), the line passes directly through the origin (0,0).

Understanding this form makes it easier to quickly identify these key characteristics of a linear equation and simplifies the process of graphing.

Plotting Points

Visualizing a linear equation through graphing often begins with plotting points. This method involves finding various (x, y) pairs that satisfy the given equation and marking them on graph paper or a grid.
To plot points:

• Start by selecting different x-values, which are any numbers you want to test in the equation.
• For each chosen x-value, compute the corresponding y-value using the equation.
• Write down each pair as coordinates \((x, y)\).
• Mark these points on the graph, ensuring that you locate each x-value on the horizontal axis and the corresponding y-value on the vertical axis.

Once all the points are plotted, connect them with a straight line to illustrate the linear relationship. In our function \( f(x) = -\frac{2}{3}x \), the points (-3, 2), (0, 0), and (3, -2) reflect how the negative slope creates a line that slants downwards from left to right.