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Given f (x)= 2/3x - 1, complete Parts A and B.

Part A: Using the table provided, create five points to demonstrate that for the function f (x)= 2/3x - 1, there is exactly one output value for each corresponding input value. In your final answer, include all calculations and the completed table.









Part B: On a separate sheet of paper, use the points created in Part A to graph the function f (x)= 2/3x - 1. Label the values on the
- and
-axes, and all points on the graph.

3 answers

Let's start with Part A, where we will create five points for the function \( f(x) = \frac{2}{3}x - 1 \). We will select five different values for \( x \), substitute them into the function, and calculate their corresponding \( f(x) \) values.

Step 1: Choose Input Values

Let’s choose the following input values for \( x \):

  • \( x = -3 \)
  • \( x = -1 \)
  • \( x = 0 \)
  • \( x = 3 \)
  • \( x = 6 \)

Step 2: Calculate Corresponding Output Values

Now we will compute \( f(x) \) for each of these values.

  1. For \( x = -3 \) \[ f(-3) = \frac{2}{3}(-3) - 1 = -2 - 1 = -3 \]
  2. For \( x = -1 \) \[ f(-1) = \frac{2}{3}(-1) - 1 = -\frac{2}{3} - 1 = -\frac{2}{3} - \frac{3}{3} = -\frac{5}{3} \approx -1.67 \]
  3. For \( x = 0 \) \[ f(0) = \frac{2}{3}(0) - 1 = 0 - 1 = -1 \]
  4. For \( x = 3 \) \[ f(3) = \frac{2}{3}(3) - 1 = 2 - 1 = 1 \]
  5. For \( x = 6 \) \[ f(6) = \frac{2}{3}(6) - 1 = 4 - 1 = 3 \]

Step 3: Create the Completed Table

Now, we can summarize the \( x \) values and corresponding \( f(x) \) values in a table:

| \( x \) | \( f(x) \) | |---------|----------------| | -3 | -3 | | -1 | -\(\frac{5}{3}\) (approx -1.67) | | 0 | -1 | | 3 | 1 | | 6 | 3 |

Conclusion for Part A

The table demonstrates that for each input \( x \), there is exactly one output \( f(x) \), indicating that \( f(x) = \frac{2}{3}x - 1 \) is a function.

Part B: Graphing the Function

Now we will use the points created in Part A to graph the function \( f(x) = \frac{2}{3}x - 1 \). Below are the points we will plot:

  • \( (-3, -3) \)
  • \( (-1, -\frac{5}{3}) \) or approximately \( (-1, -1.67) \)
  • \( (0, -1) \)
  • \( (3, 1) \)
  • \( (6, 3) \)

Step 1: Draw the Axes

  • Draw the horizontal axis (x-axis) and label it.
  • Draw the vertical axis (y-axis) and label it.

Step 2: Plot the Points

Mark the following points on the graph:

  • Point (-3, -3)
  • Point (-1, -1.67) (approximately between -1 and -2 on the y-axis)
  • Point (0, -1)
  • Point (3, 1)
  • Point (6, 3)

Step 3: Draw the Line

Connect the points with a straight line through them since this is a linear function.

Step 4: Label the Graph

  • Label each point on the graph with its coordinates.
  • Label the axes appropriately (\( x \) and \( y \)).

Graphing Note

Since I cannot create visual images, kindly take these instructions to draw the graph on a separate piece of paper as described, using a coordinate plane with adequately scaled axes. The function slopes upwards as expected with a positive slope of \( \frac{2}{3} \) and a y-intercept at -1.

can you make it less advanced

Sure! Let's break it down into simpler steps.

Part A: Finding Points

We need to choose some numbers (inputs) for \( x \), plug them into the function \( f(x) = \frac{2}{3}x - 1 \), and find the outputs. Here are five simple values for \( x \):

  1. \( x = -3 \)
  2. \( x = -1 \)
  3. \( x = 0 \)
  4. \( x = 3 \)
  5. \( x = 6 \)

Now let's calculate \( f(x) \) for each \( x \):

  1. For \( x = -3 \): \[ f(-3) = \frac{2}{3}(-3) - 1 = -2 - 1 = -3 \]

  2. For \( x = -1 \): \[ f(-1) = \frac{2}{3}(-1) - 1 = -\frac{2}{3} - 1 = -\frac{2}{3} - \frac{3}{3} = -\frac{5}{3} \approx -1.67 \]

  3. For \( x = 0 \): \[ f(0) = \frac{2}{3}(0) - 1 = 0 - 1 = -1 \]

  4. For \( x = 3 \): \[ f(3) = \frac{2}{3}(3) - 1 = 2 - 1 = 1 \]

  5. For \( x = 6 \): \[ f(6) = \frac{2}{3}(6) - 1 = 4 - 1 = 3 \]

Table of Points

Now we put these points into a table:

| \( x \) | \( f(x) \) | |---------|--------------| | -3 | -3 | | -1 | -1.67 (or -\(\frac{5}{3}\)) | | 0 | -1 | | 3 | 1 | | 6 | 3 |

Conclusion for Part A

In this table, for each \( x \) value, there is exactly one \( f(x) \) value. This shows the function works as expected.

Part B: Graphing the Function

Now let's graph the function using the points from the table. Here’s a simple guide:

  1. Draw the Axes:

    • Draw a horizontal line (x-axis).
    • Draw a vertical line (y-axis).
    • Mark where they cross (this is called the origin, (0,0)).

  2. Mark Points:

    • Plot each point we found:
      • (-3, -3): Go left to -3 on the x-axis and down to -3 on the y-axis.
      • (-1, -1.67): Go left to -1 on the x-axis and a bit below -1 on the y-axis (about halfway to -2).
      • (0, -1): Stay at the origin (0 on the x-axis) and go down to -1 on the y-axis.
      • (3, 1): Go right to 3 on the x-axis and up to 1 on the y-axis.
      • (6, 3): Go right to 6 on the x-axis and up to 3 on the y-axis.

  3. Draw the Line:

    • After you have all the points, use a ruler to draw a straight line through them because this function is linear.

  4. Label the Points:

    • Write down the coordinates next to each point for clarity.

Summary

Now you have a calculated table and a graph! For every \( x \), there is one \( f(x) \), and the graph should show a straight line connecting all the points. Enjoy graphing!