Problem 24 For the following exercises, set... [FREE SOLUTION]

Chapter 1: Problem 24

For the following exercises, set up a table to sketch the graph of each function using the following values: $x=-3,-2,-1,0,1,2,3$ $$ f(x)=\frac{1}{2} x+1 $$

Short Answer

Expert verified

Create a table, substitute the x-values into the function, compute f(x) for each x, and plot the points.

Step by step solution

Identify the Function

The function given in the exercise is a linear function: $f(x) = \frac{1}{2} x + 1$. It indicates a straight line with a slope of $\frac{1}{2}$ and a y-intercept at $y=1$.

Create a Table of Values

Create a table with two columns, one for $x$ values and one for $f(x)$. We will plug in the given $x$ values of $-3, -2, -1, 0, 1, 2, 3$ into the function $f(x) = \frac{1}{2} x + 1 $ to find the corresponding $f(x)$ values.

Calculate Each f(x) Value

- For $x = -3$: $f(-3) = \frac{1}{2}(-3) + 1 = -\frac{3}{2} + 1 = -\frac{1}{2}$- For $x = -2$: $f(-2) = \frac{1}{2}(-2) + 1 = -1 + 1 = 0$- For $x = -1$: $f(-1) = \frac{1}{2}(-1) + 1 = -\frac{1}{2} + 1 = \frac{1}{2}$- For $x = 0$: $f(0) = \frac{1}{2}(0) + 1 = 1$- For $x = 1$: $f(1) = \frac{1}{2}(1) + 1 = \frac{1}{2} + 1 = \frac{3}{2}$- For $x = 2$: $f(2) = \frac{1}{2}(2) + 1 = 1 + 1 = 2$- For $x = 3$: $f(3) = \frac{1}{2}(3) + 1 = \frac{3}{2} + 1 = \frac{5}{2}$

Complete the Table

Use the calculated $f(x)$ values to complete the table:\[\begin{array}{|c|c|}\hlinex & f(x) \\hline-3 & -\frac{1}{2} \-2 & 0 \-1 & \frac{1}{2} \0 & 1 \1 & \frac{3}{2} \2 & 2 \3 & \frac{5}{2} \\hline\end{array}\]

Sketch the Graph

Using the table of values, plot each $(x, f(x))$ point on the Cartesian plane. Connect the points with a straight line since $f(x)$ is a linear function. The points should form a line passing through the y-intercept at $y = 1$ and having a consistent slope of $\frac{1}{2}$.

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Key Concepts

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

Table of Values

To understand how a linear function behaves, starting with a table of values is essential. This table is simply a collection of particular x-values and their corresponding y-values, which we get from plugging x into the function. For the function $f(x) = \frac{1}{2} x + 1$, the x-values given are $-3, -2, -1, 0, 1, 2, 3$. These values are like stepping stones that help us predict the graph's path.
Creating the table is straightforward:

Pick an x-value from the list provided.
Substitute it into the function $f(x)$.
Calculate the corresponding f(x) value.
Record this pair (x, f(x)) in the table.

Repeating these steps for all the given x-values completes the table. This systematic approach ensures that you have all necessary coordinates to move to the next step: graph sketching.

Graph Sketching

Once the table of values is ready, graph sketching becomes a more visual task. It's the process of plotting these points on the Cartesian plane and connecting them to reveal the shape of our function. With linear functions like $f(x) = \frac{1}{2} x + 1$, the points will align in a straight path.
Start by drawing your axes and marking each x-value along the horizontal line. For each x, plot the point paired with its f(x) value from your table. After plotting all points, draw a straight line through them.

Ensure the line extends across the axes to visualize the function's continuous nature.
Check that the line touches the y-axis at the y-intercept $y=1$, confirming your calculations were accurate.

Keeps in mind that graph sketching isn't about artistic precision鈥攊t's about understanding how variables relate and change in the function.

Slope and Intercept

The slope and intercept of a linear function tell us a lot about the line represented by the function. In $f(x) = \frac{1}{2} x + 1$, the slope is $\frac{1}{2}$, and the y-intercept is 1.
**Slope ($\frac{1}{2}$)**:

The slope determines how steep the line is. A slope of $\frac{1}{2}$ means that for every 2 units you move horizontally (to the right), the line moves 1 unit up.
A positive slope, as seen here, indicates the line rises as it moves from left to right.

**Y-Intercept (1)**:

This is the point where the line crosses the y-axis. For this function, it crosses at 1.
It's also the value of $f(x)$ when $x = 0$, signifying the starting point of the line on the vertical axis.

Understanding these components helps you predict and verify the line's path, linking algebraic expressions to their geometric interpretations.

91影视

For the following exercises, set up a table to sketch the graph of each function using the following values: \(x=-3,-2,-1,0,1,2,3\) $$ f(x)=\frac{1}{2} x+1 $$

Short Answer

Step by step solution

Identify the Function

Create a Table of Values

Calculate Each f(x) Value

Complete the Table

Sketch the Graph

Key Concepts

Table of Values

Graph Sketching

Slope and Intercept

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