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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.

If f(x) = 2x^2 - 4, give the function g which shows the graph of f(x) after: (a) translation by (1,0) followed by a horizontal stretch of scale factor 1/4.

Learn how to transform the quadratic function f(x) = 2x^2 - 4 by applying a horizontal translation by 1 unit and a horizontal stretch with a scale factor of 1/4. The process involves key concepts in function transformations, such as translations and stretches. This problem is suitable for students in Grades 10-12.

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