https://file.aiquickdraw.com/mathbot/file/1726403762955axte29m8.jpg

Let's focus on solving part **d**, which asks for the equation of the resulting graph \( g(x) \) when:

\[ f(x) = 2x^2 - x + 4 \]
is **horizontally stretched with scale factor 3**.

### Horizontal Stretch:
A horizontal stretch with scale factor \( k \) affects the function by replacing every \( x \) with \( \frac{x}{k} \). For a scale factor of 3, this results in replacing \( x \) with \( \frac{x}{3} \).

So, the transformation becomes:
\[
g(x) = f\left(\frac{x}{3}\right)
\]

### Applying the Transformation:
Substitute \( \frac{x}{3} \) into the original function \( f(x) = 2x^2 - x + 4 \):

\[
g(x) = 2\left(\frac{x}{3}\right)^2 - \left(\frac{x}{3}\right) + 4
\]

### Simplifying the Expression:
1. \( \left(\frac{x}{3}\right)^2 = \frac{x^2}{9} \)
2. Substitute back into the equation:

\[
g(x) = 2 \cdot \frac{x^2}{9} - \frac{x}{3} + 4
\]

\[
g(x) = \frac{2x^2}{9} - \frac{x}{3} + 4
\]

Thus, the equation of the transformed function is:
\[
g(x) = \frac{2x^2}{9} - \frac{x}{3} + 4
\]

Would you like more details or have any questions about this transformation?

Here are 5 related questions to deepen your understanding:
1. What happens when a graph is stretched horizontally by a scale factor smaller than 1?
2. How does a horizontal compression differ from a horizontal stretch?
3. How would the equation change if the stretch factor was negative?
4. What is the effect of a vertical stretch on the function \( f(x) \)?
5. How does the vertex of a quadratic function move during horizontal stretches?

**Tip:** Horizontal transformations are applied to the \( x \)-variable, while vertical transformations affect the whole function output.

f(x) = 2x^2 - x + 4 is horizontally stretched with scale factor 3.

This problem demonstrates how to horizontally stretch the quadratic function f(x) = 2x^2 - x + 4 by a scale factor of 3. The solution covers the concept of horizontal stretching and how it affects the equation, providing a detailed step-by-step explanation.

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