Problem 7 - College Algebra with Modeling and Visualization

Question

Graph \(y=f(x)\) by hand by first plotting points to determine the shape of the graph. $$ f(x)=2 x $$

Step-by-Step Solution

Answer

Plot points (-1, -2), (0, 0), and (1, 2), then draw a line through them.

Step 1: Select x-values

Choose a few x-values to plug into the function to find corresponding y-values. A good range for linear functions is typically at least three values spanning both positive and negative numbers, such as -1, 0, and 1.

Step 2: Calculate y-values

Substitute the chosen x-values into the function to find y-values. For example, for x = -1, \( f(-1) = 2(-1) = -2 \).For x = 0, \( f(0) = 2(0) = 0 \).For x = 1, \( f(1) = 2(1) = 2 \).

Step 3: Plot Points on the Coordinate Plane

Using the calculated pairs (-1, -2), (0, 0), and (1, 2), plot these as points on a Cartesian coordinate plane.

Step 4: Draw the Graph

Connect the plotted points with a straight line, as the function is linear. Extend the line in both directions, adding arrowheads to indicate that the line continues indefinitely.

Step 5: Label the Graph

Label the graph with the function's equation, \( y = 2x \), to identify it clearly. Additionally, you might label the x and y axes.

Key Concepts

Linear Equations

A linear equation is a mathematical statement that represents a straight line when plotted on a graph. It has the form \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept. For the function \( f(x) = 2x \), the equation is already in linear form. Here, the slope \( m \) is 2, and the y-intercept \( b \) is 0. This means that for every unit increase in \( x \), \( y \) increases by 2.
Understanding the components of a linear equation is key:

• Slope (\( m \)): Indicates the steepness of the line. A positive slope means the line ascends from left to right, as in our function \( f(x) = 2x \).
• y-intercept (\( b \)): The point where the line crosses the y-axis. In \( f(x) = 2x \), the line passes through the origin (0,0).

With this knowledge, you can determine how any linear equation will appear when graphed.

Coordinate Plane

The coordinate plane, known as the Cartesian plane, is a two-dimensional space where any point can be represented by a pair of numbers (x, y). It consists of two axes:

• x-axis: A horizontal line that runs left to right.
• y-axis: A vertical line that runs up and down.

These axes divide the plane into four quadrants, each providing a unique setting for plotting points.
In the exercise where we graphed \( y = 2x \), we used the coordinate plane to visualize our points. By plotting points and connecting them, we create a graph that illustrates the behavior of the function.
Using a coordinate plane allows us to easily see how changes in \( x \) affect \( y \), and thus how a linear equation like \( f(x) = 2x \) is structured.

Plotting Points

Plotting points is a crucial step in graphing linear functions. Here's the basic process:

• Select a few x-values. For linear equations, choosing at least three values helps establish a clear line.
• Calculate the corresponding y-values by substituting the x-values into the function. This gives you set "coordinates" or pairs \((x, y)\).
• Mark these points on the coordinate plane.

For this exercise, we chose x-values (-1, 0, 1). Calculating the y-values gave us points (-1, -2), (0, 0), and (1, 2).
Once these points are plotted, draw a line through them. Make sure the line is straight because, for linear functions, points form a straight line. Extend it with arrowheads to imply that it continues indefinitely. This technique demonstrates how plotting points directly helps graph the function \( y = 2x \) accurately.