Solutions:

Find the value of $f(5)$ for each function:

13. $f(x) = 6 + 3x$
$f(5) = 6 + 3(5) = 6 + 15 = 21$

14. $f(x) = -2(x + 1)$
$f(5) = -2(5 + 1) = -2(6) = -12$

15. $f(a) = 3(a + 2) - 1$
$f(5) = 3(5 + 2) - 1 = 3(7) - 1 = 21 - 1 = 20$

16. $f(h) = -\frac{h}{10}$
$f(5) = -\frac{5}{10} = -0.5$

17. $f(m) = 1 - 4\left(\frac{m}{2}\right)$
$f(5) = 1 - 4\left(\frac{5}{2}\right) = 1 - 4(2.5) = 1 - 10 = -9$

18. $f(m) = 2(m - 3)$
$f(5) = 2(5 - 3) = 2(2) = 4$

Write a linear function for the data in each table:

19. Table:
To find the linear function $y = mx + b$:

• Slope: $m = \frac{\Delta y}{\Delta x} = \frac{4 - (-1)}{1 - 0} = 5$
• Using point $(0, -1)$, $b = -1$.

$y = 5x - 1$

20. Table:
Slope $m = \frac{\Delta y}{\Delta x} = \frac{1.5 - 4}{1 - 0} = -2.5$
Using $(0, 4)$, $b = 4$:

$y = -2.5x + 4$

21. Table:
Slope $m = \frac{\Delta y}{\Delta x} = \frac{1/2 - 2}{-1 - (-2)} = -\frac{3}{2}$
Using $(-2, 2)$, $b = 2 - \left(-\frac{3}{2} \cdot -2\right) = 2 - 3 = -1$:

$y = -\frac{3}{2}x - 1$

Sketch the graph of each linear function:

22. $g(x) = x - 3$
This is a line with slope 1 and $y$-intercept at -3.

23. $h(x) = 3 - x$
This is a line with slope -1 and $y$-intercept at 3.

Let me know if you want me to provide detailed graphs, additional explanations, or solve further!

Follow-up Questions:

1. How do you calculate the slope from a table of values?
2. What is the significance of $b$ in the equation $y = mx + b$?
3. How do negative slopes affect the direction of a graph?
4. Can you provide examples of nonlinear functions and how they differ graphically?
5. Why is it important to plug in values to verify solutions?

Tip: Always label your axes and points clearly when graphing for better visualization and accuracy.