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To find the range of the quadratic function \( f(x) = 2x^2 + 4x - 1 \), follow these steps:

### Step 1: Identify the vertex
The range of a quadratic function is determined by the vertex. The general form of a quadratic function is:

\[
f(x) = ax^2 + bx + c
\]

For this function:
- \( a = 2 \)
- \( b = 4 \)
- \( c = -1 \)

The x-coordinate of the vertex is given by the formula:

\[
x_{\text{vertex}} = -\frac{b}{2a}
\]

Substituting the values of \( a \) and \( b \):

\[
x_{\text{vertex}} = -\frac{4}{2(2)} = -\frac{4}{4} = -1
\]

### Step 2: Find the y-coordinate of the vertex
Now, substitute \( x = -1 \) back into the function to find the y-coordinate:

\[
f(-1) = 2(-1)^2 + 4(-1) - 1
\]

\[
f(-1) = 2(1) - 4 - 1 = 2 - 4 - 1 = -3
\]

Thus, the vertex is at \( (-1, -3) \).

### Step 3: Determine the range
Since \( a = 2 \), which is positive, the parabola opens upwards. This means that the function's values will increase without bound above the vertex. The lowest point on the graph is \( -3 \).

Thus, the range of the function is:

\[
[-3, \infty)
\]

### Final Answer:
In interval notation, the range is \([ -3, \infty )\).

Would you like further details or explanations on this?

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Here are five related questions you might consider:
1. How do you determine the direction a parabola opens based on the coefficient of \(x^2\)?
2. What is the general formula for finding the vertex of a quadratic function?
3. How would the range change if the quadratic coefficient were negative?
4. What are the effects of shifting the parabola vertically or horizontally on its range?
5. Can the range of a quadratic function ever be all real numbers?

**Tip:** Always check the sign of the leading coefficient \(a\) in a quadratic equation to know if the parabola opens upwards or downwards, as this will directly affect the range.

Find the range of the quadratic function f(x) = 2x^2 + 4x - 1. Write your answer using interval notation.

Learn how to find the range of the quadratic function f(x) = 2x^2 + 4x - 1 using the vertex formula. This step-by-step guide explains how to calculate the vertex and the range in interval notation.

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