Problem 27 Graph each function by making a ... [FREE SOLUTION]

Chapter 4: Problem 27

Graph each function by making a table of values and plotting points. $$f(x)=\frac{2}{3} x+2$$

Short Answer

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To graph the function $f(x) = \frac{2}{3}x + 2$, first create a table of values for different x values: | x | y | |-----|--------| | -3 | 0 | | -2 | 2/3 | | -1 | 4/3 | | 0 | 2 | | 1 | 8/3 | | 2 | 10/3 | | 3 | 4 | Next, plot these points on a coordinate plane and connect them to form a straight line, representing the graph of the function $f(x) =\frac{2}{3}x+2$.

Step by step solution

Choose the x values

Choose some integer values of x to find the corresponding y values (function values). Let's choose -3, -2, -1, 0, 1, 2, and 3 as our x values.

Calculate the y values

Calculate the y values (function values) by plugging each x value we chose in Step 1 into the function $f(x) = \frac{2}{3}x + 2$. f(-3) = $\frac{2}{3}(-3) + 2 = -2 + 2 = 0$ f(-2) = $\frac{2}{3}(-2) + 2 = -\frac{4}{3} + 2 = \frac{2}{3}$ f(-1) = $\frac{2}{3}(-1) + 2 = -\frac{2}{3} + 2 = \frac{4}{3}$ f(0) = $\frac{2}{3}(0) + 2 = 0 + 2 = 2$ f(1) = $\frac{2}{3}(1) + 2 = \frac{2}{3} + 2 = \frac{8}{3}$ f(2) = $\frac{2}{3}(2) + 2 = \frac{4}{3} + 2 = \frac{10}{3}$ f(3) = $\frac{2}{3}(3) + 2 = 2 + 2 = 4$

Create the table of values

Create a table of values for our selected x values and their corresponding y values (function values) obtained in Step 2: | x | y | |-----|--------| | -3 | 0 | | -2 | 2/3 | | -1 | 4/3 | | 0 | 2 | | 1 | 8/3 | | 2 | 10/3 | | 3 | 4 |

Plot the points

Plot the points from the table of values on a coordinate plane: | x | y | |-----|--------| | -3 | 0 | | -2 | 2/3 | | -1 | 4/3 | | 0 | 2 | | 1 | 8/3 | | 2 | 10/3 | | 3 | 4 |

Connect the points to form a line

Connect the points plotted in Step 4 to form a straight line. This line represents the graph of the function $f(x) = \frac{2}{3}x + 2$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Table of Values

Creating a table of values is a great way to start graphing a linear function. It helps in finding several points that satisfy the function by choosing specific values for x, and calculating the corresponding y values. Here's how you can do it:

First, select a set of x values. It is often helpful to choose a variety of negative, zero, and positive values to see a more complete picture of the line.
Plug these x values into the function equation to find the y values.
Record these pairs (x, y) in a table. Each pair will later become a point on your graph.

This method gives a clear view of how the function behaves and is especially helpful for beginners. In our example, we used the function $f(x) = \frac{2}{3}x + 2$. For each selected x value, we calculated y and filled our table of values.

Plotting Points

Once you have a table of values, the next step is plotting those points on a graph. This step brings the numbers in your table to life in the form of a visual representation of the function. Here's how to plot points:

Find the point on the graph where your first x value matches with the y value from your table.
Make a small dot at that location.
Repeat this process for each pair of numbers from your table.
Ensure that you have even spacing and that each point is marked clearly.

Once all points are plotted, you can start to see the shape and direction of your graph. This visual step is crucial because it sets the stage for drawing the next line, making sure your graph accurately represents the function you are working with.

Coordinate Plane

Understanding the coordinate plane is essential when graphing linear functions. The coordinate plane consists of two axes: the horizontal x-axis and the vertical y-axis. Each point you plot corresponds to an ordered pair (x, y). Here's why the coordinate plane matters:

The x-axis is where you'll find the x values from your table.
The y-axis represents the y values you've calculated.
By plotting where these values intersect, you can draw a line that represents the function.
Remember that each quadrant of the plane can show different parts of your line, so always consider both negative and positive values.

The plane provides a clear layout for visualizing mathematical relationships and is a fundamental tool for graphing. Once you have plotted all the points from your table, you simply connect these points with a line, giving you the complete graph of the linear function.

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91影视

Graph each function by making a table of values and plotting points. $$f(x)=\frac{2}{3} x+2$$

Short Answer

Step by step solution

Choose the x values

Calculate the y values

Create the table of values

Plot the points

Connect the points to form a line

Key Concepts

Table of Values

Plotting Points

Coordinate Plane

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