The function you have provided is:

\[
f(x) = 2x^2
\]

This is a quadratic function, where \( a = 2 \), and it has the general form \( f(x) = ax^2 + bx + c \), with \( b = 0 \) and \( c = 0 \). Let's go over some key properties of this function:

1. **Vertex**: Since there is no linear term (\(bx\)) or constant term (\(c\)), the vertex is at the origin, \((0,0)\).
   
2. **Axis of symmetry**: The axis of symmetry is the vertical line \( x = 0 \), which passes through the vertex.

3. **Direction of the parabola**: The coefficient of \(x^2\) is positive, so the parabola opens **upward**.

4. **Domain**: The function is defined for all real values of \(x\), so the domain is \( (-\infty, \infty) \).

5. **Range**: Since the parabola opens upward and the minimum value occurs at \( f(0) = 0 \), the range is \( [0, \infty) \).

6. **Growth rate**: As \(x\) increases or decreases away from 0, the value of \(f(x)\) grows quadratically, meaning it grows faster than a linear function.

Would you like more details or any specific questions answered?

Here are 5 related questions:
1. How would the graph change if the coefficient of \(x^2\) was negative?
2. What are the intercepts of this quadratic function?
3. How can the derivative of this function help describe its behavior?
4. How does this quadratic compare to \(f(x) = x^2\)?
5. Can you express this function in vertex form, even though the vertex is already at the origin?

**Tip**: The coefficient of \(x^2\) controls how "wide" or "narrow" the parabola is—larger values make the graph narrower.

Explore the quadratic function f(x) = 2x^2, including its vertex, axis of symmetry, direction of parabola, domain, range, and growth rate. Understand how the function behaves and how it compares to other quadratic functions. Suitable for students in Grades 8-10.

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