To identify the transformations that represent a vertical stretch of the parent function $f(x) = x$, we need to focus on the coefficient of $x$, which is the value that multiplies $x$. A vertical stretch occurs when this coefficient is greater than 1 or less than -1 (i.e., its absolute value is greater than 1).

Let's analyze each option:

• A. $y = \frac{2}{3}x - 1$: The coefficient of $x$ is $\frac{2}{3}$, which is less than 1. This represents a vertical compression, not a stretch.

• B. $y = x + 5$: The coefficient of $x$ is 1. This is the original parent function with a vertical shift, not a stretch.

• C. $y = \frac{5}{2}x + 6$: The coefficient of $x$ is $\frac{5}{2}$, which is greater than 1. This represents a vertical stretch.

• D. $y = 0.75x - 9$: The coefficient of $x$ is 0.75, which is less than 1. This is a vertical compression, not a stretch.

• E. $y = 4x - 2$: The coefficient of $x$ is 4, which is greater than 1. This represents a vertical stretch.

Thus, the correct answers are:

• C. $y = \frac{5}{2}x + 6$
• E. $y = 4x - 2$

Would you like more details or have any further questions?

Here are five related questions to deepen understanding:

1. What is the difference between vertical stretching and compression?
2. How do vertical shifts affect the graph of a function?
3. Can you describe a horizontal stretch, and how does it differ from vertical stretching?
4. How does the transformation $y = -kx$ differ from $y = kx$ in terms of graph behavior?
5. What happens to the graph if a negative coefficient is used for $x$ (e.g., $y = -2x$)?

Tip: To recognize vertical stretching, always check the absolute value of the coefficient of $x$. If it’s greater than 1, the graph stretches vertically.