Here is a breakdown of the questions and solutions for slope, rate of change, and graph transformations from the three uploaded images.

Part 1: Slope and Rate of Change

Example 1: Find the slope of the line through points $(1, 7)$ and $(2, -4)$.

The slope formula is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute $(x_1, y_1) = (1, 7)$ and $(x_2, y_2) = (2, -4)$: $m = \frac{-4 - 7}{2 - 1} = \frac{-11}{1} = -11$

The slope is -11.

Example 2: Find the slope of $y = \frac{2}{3}x - 4$.

The equation is already in slope-intercept form $y = mx + b$, where $m$ is the slope.

Here, $m = \frac{2}{3}$.

The slope is $\frac{2}{3}$.

Example 3:

The graph represents the growth of a plant over 20 weeks.

(A) Find the average growth rate between 5 weeks and 10 weeks.

To calculate the average rate of change: $\text{Rate of Change} = \frac{\Delta y}{\Delta x}$ From the graph:

• At 5 weeks, the length is approximately 12 cm.
• At 10 weeks, the length is approximately 36 cm.

$\text{Rate} = \frac{36 - 12}{10 - 5} = \frac{24}{5} = 4.8 \, \text{cm/week}.$

(B) Find the average growth rate between 13 weeks and 16 weeks.

From the graph:

• At 13 weeks, the length is approximately 40 cm.
• At 16 weeks, the length is approximately 43 cm.

$\text{Rate} = \frac{43 - 40}{16 - 13} = \frac{3}{3} = 1 \, \text{cm/week}.$

(C) Which rate of change is slowest?

The slowest rate is 1 cm/week (between 13 and 16 weeks).

Part 2: Graphing Transformations

Function 1: $y = \sqrt{x - 3}$

• Transformation: Horizontal shift 3 units right.
• The graph of $y = \sqrt{x}$ is shifted to the right by 3 units.

Function 2: $y = -|x|$

• Transformation: Reflection over the x-axis.
• The graph of $y = |x|$ flips downward.

Function 3: $y = (x + 1)^3 + 2$

• Transformation:
  • Horizontal shift 1 unit left.
  • Vertical shift 2 units up.
• The graph of $y = x^3$ shifts as described.

Unlabeled Graphs (Page 3)

1. Top Graph: A standard parabola $y = x^2$, representing a basic quadratic function.
2. Middle Graph: The V-shape indicates an absolute value function $y = |x|$.
3. Bottom Graph: The curve suggests a cubic function $y = x^3$.

Summary of Key Concepts

1. The slope formula calculates how steep a line is between two points.
2. Rate of change measures how a quantity changes relative to another.
3. Graph transformations:
   • Horizontal shifts: $x \to x - h$ (right) or $x \to x + h$ (left).
   • Vertical shifts: $y \to y + k$ (up) or $y \to y - k$ (down).
   • Reflections: Over the $x$-axis ($-f(x)$) or $y$-axis ($f(-x)$).

Do you want detailed explanations for any of these? Let me know!

Related Questions:

1. How do you calculate slope when given two points?
2. What does the slope of a real-life graph represent in practical terms?
3. How do you determine transformations from a function’s equation?
4. What is the difference between vertical and horizontal shifts?
5. How do reflections affect the graph of a function?

Tip: Always compare points carefully when calculating slope or rate of change to avoid sign errors.