Problem 47 Complete each table of function ... [FREE SOLUTION]

Chapter 3: Problem 47

Complete each table of function values and then graph each function. See Examples 5 and 6. $$ \begin{aligned} &f(x)=-3 x-2\\\ &\begin{array}{|c|c|} \hline x & {f(x)} \\ \hline-2 & {} \\ {-1} & {} \\ {0} & {} \\ {1} & {} \\ \hline \end{array} \end{aligned} $$

Short Answer

Expert verified

The table is completed: $(-2,4), (-1,1), (0,-2), (1,-5)$. Graph these points and draw the line.

Step by step solution

Understand the Function

The function given is $ f(x) = -3x - 2 $. This function is a linear function, which will produce a straight line when graphed. The slope of the line is $-3$, and the y-intercept is $-2$.

Calculate the Function Values

We will compute the function values $ f(x) $ for each provided $ x $ value. 1. When $ x = -2 $, $ f(-2) = -3(-2) - 2 = 6 - 2 = 4 $. 2. When $ x = -1 $, $ f(-1) = -3(-1) - 2 = 3 - 2 = 1 $. 3. When $ x = 0 $, $ f(0) = -3(0) - 2 = 0 - 2 = -2 $. 4. When $ x = 1 $, $ f(1) = -3(1) - 2 = -3 - 2 = -5 $.

Complete the Function Values Table

Now, we can fill the table with the computed $ f(x) $ values: \[\begin{array}{|c|c|}\hline x & f(x) \\hline-2 & 4 \-1 & 1 \0 & -2 \1 & -5 \\hline\end{array}\]

Graph the Function

Plot the calculated points $(-2,4), (-1,1), (0,-2), (1,-5)$ on the graph. Connect the points to form a straight line, which accurately represents the function $ f(x) = -3x - 2 $. The line should confirm the slope of $-3$ and y-intercept at $-2$.

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Key Concepts

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

Function Table

A function table is a simple way to organize data that represents a function, specifically a linear function in this case. To understand how a function table works, imagine it as a guide that tells you what the output of a function will be for specific inputs. For example, with the function $ f(x) = -3x - 2 $, you can determine the function values by inserting different values for $ x $ into the equation and calculating the results.

The function table helps to neatly display these calculations:

For $ x = -2 $: insert into the function, $ f(-2) = -3(-2) - 2 = 6 - 2 = 4 $.

For $ x = -1 $: $ f(-1) = -3(-1) - 2 = 3 - 2 = 1 $.

For $ x = 0 $: $ f(0) = -3(0) - 2 = 0 - 2 = -2 $.

For $ x = 1 $: $ f(1) = -3(1) - 2 = -3 - 2 = -5 $.

Each row of the table connects an $ x $ value with its corresponding $ f(x) $ result, making it easy to see how changes in $ x $ affect the outcomes of the function.

Graphing Functions

Graphing functions gives a visual representation of how a function behaves. For linear functions such as $ f(x) = -3x - 2 $, the graph will be a straight line. The benefit of graphing is that it allows you to instantly see the relationship between $ x $ and $ f(x) $.

To create the graph:

Begin by plotting the points determined from your function table onto a coordinate plane: $(-2, 4)$, $(-1, 1)$, $(0, -2)$, and $(1, -5)$.

Once these points are plotted, use a ruler to connect them with a straight line. Ensure the line extends beyond the plotted points to cover the entire range of the function.

This line reflects all possible values that $ f(x) $ can take, based on any real number value for $ x $.

By using a graph, you gain insight into the slope of the function, allowing you to quickly analyze its steepness and direction. Graphs serve as a powerful tool for understanding the nature of linear equations.

Slope and Y-intercept

The slope and y-intercept are key components of any linear function and are crucial for quickly identifying the characteristics of a function represented by a straight line. The function given, $ f(x) = -3x - 2 $, is in the form $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept.

Understanding slope:

The slope of the line, represented by $ -3 $ in the equation, indicates how steep the line is. It tells us for every 1 unit increase in $ x $, $ f(x) $ decreases by 3 units. This negative slope shows the line descends as it moves to the right.

Regarding the y-intercept:

The y-intercept, indicated by -2, shows where the line crosses the y-axis. This point is crucial as it gives a starting value when $ x = 0 $. In graphical terms, the function's line will intersect the vertical y-axis at the point $(0, -2)$.

Together, the slope and y-intercept provide a comprehensive view of the function's graph, helping both in its creation and in predicting its behavior.

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91影视

Complete each table of function values and then graph each function. See Examples 5 and 6. $$ \begin{aligned} &f(x)=-3 x-2\\\ &\begin{array}{|c|c|} \hline x & {f(x)} \\ \hline-2 & {} \\ {-1} & {} \\ {0} & {} \\ {1} & {} \\ \hline \end{array} \end{aligned} $$

Short Answer

Step by step solution

Understand the Function

Calculate the Function Values

Complete the Function Values Table

Graph the Function

Key Concepts

Function Table

Graphing Functions

Slope and Y-intercept

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