# Memory Palace for Math 105 Exam Practice 1 (Spring 2025)

## Front Door – Fraction Fun


- **Theme:** A giant door painted with pizza slices.
- **Key Concepts:**
- **Multiplying Fractions (Multiply straight across):**
Imagine the pizza slices being combined
$$\frac{3}{7} \times \left(-\frac{5}{2}\right) = -\frac{15}{14}$$
- **Subtracting Fractions (Need LCD to combine):**
Picture subtracting toppings :
$$\frac{1}{4} - \frac{2}{3} = -\frac{5}{12}$$

## Entry Hall – Order of Operations (PEMDAS)


- **Theme:** A hallway lined with colorful signposts reading "P – E – M/D – A/S."
- **Key Concept:**
Recall "Please Excuse My Dear Aunt Sally" to evaluate:
$$-4 \cdot 3^2 + 2\cdot 3 - 6$$
Follow the steps using PEMDAS.

## Living Room – Coordinate Constellation


- **Theme:** A floor grid with sparkling stars marking specific coordinates.
- **Key Concepts:**
- **Plotting Points:**
Imagine placing stars labeled:
- $A = (2,5)$
- $B = (-4,5)$
- $C = (-2,-6)$

## Kitchen – Distance Detective & Midpoint Mystery


- **Theme:** A kitchen table with a giant measuring tape and a treasure map.
- **Key Concepts:**
- **Distance Formula:**
Measure the distance between $(2,-4)$ and $(5,1)$:
$$d = \sqrt{(5-2)^2 + (1-(-4))^2} = \sqrt{34}$$
- **Midpoint Formula:**
Find the midpoint between

$(4,\frac{10}{7})$ and $(-7,-\frac{2}{5})$:
Result: $\left(-\frac{3}{2},\frac{18}{35}\right)$.

## Study/Office – Equation Check & Function Fundamentals


- **Theme:** A giant chalkboard filled with equations and relation tables.
- **Key Concepts:**
- **Checking Solutions:**
Substitute $\left(\frac{3}{2},\frac{4}{3}\right)$ into $8x - 6y = 5$ and verify the result.
- **Function Rules:**
Understand that a function assigns each input exactly one output. Visualize tables and diagrams that show no duplicate domain entries.

## Dining Room – Graphing Lines & Intercepts


- **Theme:** A dining table covered with a coordinate grid tablecloth. Tomahawk Steaks.
- **Key Concepts:**
- **Intercepts of $2x-4y=8$:**
- Plug in $y=0$ and solve for $x$ to get $x$-intercept: $(4, 0)$
- Plug in $x=0$ and solve for $y $ to get $y$-intercept: $(0, -2)$
- **Vertical Line Test:**
Picture shining a vertical flashlight across graphs to check if any vertical line hits more than one point. If it intersects a vertical line more than once, not a function. If it only intersects a vertical line at most once, it is a function.

## Basement – Domain, Slope, and Special Lines


- **Theme:** A secret laboratory with gadgets and slope diagrams on the walls. Mountain Dew and Cheetos strewn about. $9000 gaming PC.
- **Key Concepts:**
- **Domain of Rational Functions:**
For $f(x) = \frac{x+5}{7-x}$, exclude $x=7$ (since it makes the denominator zero).
- **Evaluating Functions:**
For $f(x) = \frac{x}{x-3}$:
- $f(3)$ is undefined.
- $f(6) = 2$.
- $f\left(\frac{1}{2}\right) = -\frac{1}{5}$.
- **Slope:**
Find the slope between $(2,6)$ and $(-8,-2)$:
$$m = \frac{-2-6}{-8-2} = \frac{4}{5}$$
- **Vertical Lines:**
For $x = 9$, note that the slope is undefined and there is no y-intercept.

## Backyard – Final Function Graphs


- **Theme:** A backyard mini outdoor theater showing a graph on a big screen.
- **Key Concepts:**
- **Graph Readings:**
From the graph in Figure 1:
- $f(1) = -1$
- $f(-2) = 1$
- $f(0) = 3$
- **Domain and Range:**
Identify the domain as $[-3,3]$ and the range as $[-1,3]$ in interval notation.