1. **Problem 1: Find $a - b$ for the linear function $f(x) = ax + b$ given the table values:** Given points: $(1, -64)$, $(2, 0)$, $(3, 64)$. 2. Use the formula for a linear function: $f(x) = ax + b$. 3. Substitute the points to form equations: $$ \begin{cases} a(1) + b = -64 \\ a(2) + b = 0 \\ a(3) + b = 64 \end{cases} $$ 4. From the first two equations: $$ \begin{aligned} a + b &= -64 \\ 2a + b &= 0 \end{aligned} $$ Subtract the first from the second: $$ (2a + b) - (a + b) = 0 - (-64) \\ \cancel{2a} + \cancel{b} - \cancel{a} - \cancel{b} = 64 \\ a = 64 $$ 5. Substitute $a=64$ into $a + b = -64$: $$ 64 + b = -64 \\ b = -64 - 64 = -128 $$ 6. Calculate $a - b$: $$ 64 - (-128) = 64 + 128 = 192 $$ **Answer for Problem 1:** D. 192 --- 1. **Problem 2: Find the function $f(n)$ for the museum charge for $n \geq 25$ people.** 2. The museum charges $21$ per person for the first 25 people, so total for first 25 is: $$ 25 \times 21 = 525 $$ 3. For each additional person beyond 25, charge is $14$ per person. 4. Number of additional people beyond 25 is $n - 25$. 5. Total charge function: $$ f(n) = 525 + 14(n - 25) $$ 6. Simplify: $$ f(n) = 525 + 14n - 350 = 14n + 175 $$ **Answer for Problem 2:** A. $f(n) = 14n + 175$ --- 1. **Problem 3: Find the increase in Fahrenheit temperature when Kelvin temperature increases by 9.10 kelvins.** 2. Given function: $$ F(x) = \frac{9}{5}(x - 273.15) + 32 $$ 3. Temperature increase in kelvins: $\Delta x = 9.10$ 4. Increase in Fahrenheit is: $$ \Delta F = F(x + \Delta x) - F(x) = \frac{9}{5}((x + 9.10) - 273.15) + 32 - \left(\frac{9}{5}(x - 273.15) + 32\right) $$ 5. Simplify: $$ \Delta F = \frac{9}{5}(x + 9.10 - 273.15) + 32 - \frac{9}{5}(x - 273.15) - 32 = \frac{9}{5}(9.10) = \frac{9}{5} \times 9.10 $$ 6. Calculate: $$ \frac{9}{5} \times 9.10 = 1.8 \times 9.10 = 16.38 $$ **Answer for Problem 3:** A. 16.38