Sure! Let's solve for the first function: [tex]\( f(x) = 3x + 4 \)[/tex].
### Ordered Pairs, Mapping, and Table of Values
First, we'll generate 5 integer x-values from -2 to 2.
1. Select integer x-values:
[tex]\[ x = -2, -1, 0, 1, 2 \][/tex]
2. Calculate the corresponding y-values using the equation [tex]\( f(x) = 3x + 4 \)[/tex]:
[tex]\[
\begin{align*}
f(-2) &= 3(-2) + 4 = -6 + 4 = -2 \\
f(-1) &= 3(-1) + 4 = -3 + 4 = 1 \\
f(0) &= 3(0) + 4 = 0 + 4 = 4 \\
f(1) &= 3(1) + 4 = 3 + 4 = 7 \\
f(2) &= 3(2) + 4 = 6 + 4 = 10 \\
\end{align*}
\][/tex]
3. Generate the ordered pairs:
[tex]\[
\{ (-2, -2), (-1, 1), (0, 4), (1, 7), (2, 10) \}
\][/tex]
4. Create a mapping (dictionary) from x to y:
[tex]\[
\{ -2: -2, -1: 1, 0: 4, 1: 7, 2: 10 \}
\][/tex]
5. Prepare a table of values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & -2 \\
\hline
-1 & 1 \\
\hline
0 & 4 \\
\hline
1 & 7 \\
\hline
2 & 10 \\
\hline
\end{array}
\][/tex]
### Graph of the Function
To plot the function [tex]\( f(x) = 3x + 4 \)[/tex] on graph paper:
1. Label the axes. The x-axis will range from -2 to 2, and the y-axis from -2 to 10.
2. Plot the ordered pairs:
[tex]\[
\begin{align*}
(-2, -2) \\
(-1, 1) \\
(0, 4) \\
(1, 7) \\
(2, 10) \\
\end{align*}
\][/tex]
3. Draw and connect the points. Since we have a linear equation, the points lie on a straight line.
Now, draw the line passing through these points to represent the function [tex]\( f(x) = 3x + 4 \)[/tex].
### Summary
- Ordered Pairs: \{ (-2, -2), (-1, 1), (0, 4), (1, 7), (2, 10) \}
- Mapping: \{ -2: -2, -1: 1, 0: 4, 1: 7, 2: 10 \}
- Table of Values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & -2 \\
-1 & 1 \\
0 & 4 \\
1 & 7 \\
2 & 10 \\
\hline
\end{array}
\][/tex]
- Graph:
A straight line passing through the points \{ (-2, -2), (-1, 1), (0, 4), (1, 7), (2, 10) \}, respectively.
By following these steps, you can see how the function behaves in a visual and tabular format.
