Write the equation of the parabola that has the same shape as f left parenthesis x right parenthesis equals 6 x squared but with vertex left parenthesis negative 4 comma 2 right parenthesis in the form f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k.

To write the equation of a parabola in the form \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex and \( a \) determines the shape and orientation of the parabola, we follow these steps:

1. **Identify the given information:**
   - The shape of the parabola is the same as \( f(x) = 6x^2 \). This means that the value of \( a \) is 6.
   - The vertex of the parabola is \( (-4, 2) \), so \( h = -4 \) and \( k = 2 \).

2. **Substitute \( a \), \( h \), and \( k \) into the general vertex form equation:**
   \[
   f(x) = a(x - h)^2 + k
   \]
   Substituting the given values:
   \[
   f(x) = 6(x - (-4))^2 + 2
   \]

3. **Simplify the equation:**
   \[
   f(x) = 6(x + 4)^2 + 2
   \]

So, the equation of the parabola is:

\[
f(x) = 6(x + 4)^2 + 2
\]

Would you like more details or have any questions?

Here are some related questions:

1. How do you determine the value of \( a \) for a parabola?
2. What is the effect of changing the value of \( h \) in the vertex form of a parabola?
3. How does the value of \( k \) affect the position of the parabola on the graph?
4. What are the steps to convert a quadratic equation in standard form to vertex form?
5. How do you find the axis of symmetry for a parabola given its vertex form?

**Tip:** The value of \( a \) determines the "width" and direction of the parabola. If \( a \) is positive, the parabola opens upwards; if \( a \) is negative, it opens downwards.

Learn how to write the equation of a parabola that has the same shape as f(x) = 6x^2 but with vertex (-4, 2) in the form f(x) = a(x - h)^2 + k. This problem covers quadratic functions and the vertex form of parabolas. Suitable for Grades 10-12 students.

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