A vertical stretch occurs when the graph of a function is stretched away from the x-axis. Mathematically, this happens when the coefficient of $x$ is multiplied by a factor greater than 1. A vertical compression, on the other hand, happens when this coefficient is multiplied by a factor between 0 and 1.

The parent function given is $f(x) = x$, and to vertically stretch the function, we multiply $x$ by a constant greater than 1.

Let's evaluate the choices:

• A: $y = 23x - 1$
  The coefficient of $x$ is 23, which is greater than 1. Therefore, this represents a vertical stretch.

• B: $y = x + 5$
  The coefficient of $x$ is 1, which does not result in a vertical stretch.

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

• D: $y = 0.75x - 9$
  The coefficient of $x$ is 0.75, which is less than 1, so this represents a vertical compression.

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

Correct Answers:

• A: $y = 23x - 1$
• C: $y = \frac{5}{2}x + 6$

Would you like further clarification or details on vertical transformations?
Here are some additional questions to expand your understanding:

1. What is the difference between a vertical stretch and a vertical compression?
2. How does adding or subtracting a constant (like in $y = x + 5$) affect the graph of a function?
3. What is the effect of a negative coefficient on the transformation of a function?
4. How can you identify horizontal stretches or compressions in a function?
5. How would you graph a vertically stretched function like $y = 4x$?

Tip: Always look at the coefficient of $x$ to determine whether a function is being stretched or compressed vertically.