We are given the function $f(x) = 2x^2 - 4$ and are asked to find a new function $g(x)$ that represents the graph of $f(x)$ after the following transformations:

1. Translation by $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ (right by 1 unit).
2. Horizontal stretch by a factor of $\frac{1}{4}$.

Step 1: Translation by $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$

A translation to the right by 1 unit is applied to the $x$-coordinates. This changes the function from $f(x)$ to $f(x-1)$.

So, the translated function is: $f(x-1) = 2(x-1)^2 - 4$ Now, let's expand $(x-1)^2$: $(x-1)^2 = x^2 - 2x + 1$ Thus, the translated function becomes: $f(x-1) = 2(x^2 - 2x + 1) - 4 = 2x^2 - 4x + 2 - 4 = 2x^2 - 4x - 2$

Step 2: Horizontal Stretch by a Factor of $\frac{1}{4}$

To apply a horizontal stretch by a factor of $\frac{1}{4}$, we replace $x$ in the function with $\frac{x}{\frac{1}{4}}$ (i.e., multiply $x$ by 4).

The function after this transformation is: $f\left( \frac{x}{4} - 1 \right) = 2\left( \frac{x}{4} - 1 \right)^2 - 4$ Now expand $\left( \frac{x}{4} - 1 \right)^2$: $\left( \frac{x}{4} - 1 \right)^2 = \left( \frac{x^2}{16} - \frac{x}{2} + 1 \right)$ Substitute this back into the function: $f\left( \frac{x}{4} - 1 \right) = 2 \left( \frac{x^2}{16} - \frac{x}{2} + 1 \right) - 4$ Simplify the expression: $= \frac{x^2}{8} - x + 2 - 4 = \frac{x^2}{8} - x - 2$

Thus, the function $g(x)$, after both transformations, is: $g(x) = \frac{x^2}{8} - x - 2$

Would you like more details on any of the steps?

Here are some related questions for further exploration:

1. How would the graph change if we performed a vertical stretch instead of a horizontal one?
2. What happens to the vertex of the parabola after the transformations?
3. Can we apply a reflection instead of a stretch and find the new function?
4. How does the horizontal stretch factor of $\frac{1}{4}$ affect the graph's shape?
5. What is the general process for applying multiple transformations to a function?

Tip: When applying transformations, always handle translations and stretches separately and in the correct order to avoid confusion.