Use graphing technology to find the range of the function f, of, x, equals, x, squared, plus, 6, x, plus, 1, .*f*(*x*)=*x^2*+6*x*+1.

To find the range of the function \( f(x) = x^2 + 6x + 1 \) using graphing technology, let's first analyze the function and then describe the graph.

### Analyzing the Function
The function \( f(x) = x^2 + 6x + 1 \) is a quadratic function in standard form \( ax^2 + bx + c \), where \( a = 1 \), \( b = 6 \), and \( c = 1 \).

#### Vertex of the Parabola
The vertex form of a quadratic function can be derived by completing the square or using the vertex formula:

\[
x_{\text{vertex}} = -\frac{b}{2a} = -\frac{6}{2(1)} = -3
\]

To find the corresponding \( y \) value at this \( x \):

\[
f(-3) = (-3)^2 + 6(-3) + 1 = 9 - 18 + 1 = -8
\]

So, the vertex of the parabola is at \( (-3, -8) \).

### Graphing the Function
The function \( f(x) = x^2 + 6x + 1 \) is a parabola that opens upwards (since the coefficient of \( x^2 \) is positive). The vertex is at \( (-3, -8) \), which is the minimum point on the graph.

### Determining the Range
Since the parabola opens upwards and the lowest point is at \( y = -8 \), the range of \( f(x) \) is all values of \( y \) that are greater than or equal to \( -8 \).

### Conclusion
Thus, the range of the function \( f(x) = x^2 + 6x + 1 \) is:
\[
\text{Range} = [-8, \infty)
\]

Would you like more details on the graph or have any questions?

Here are some related questions:
1. What is the domain of the function \( f(x) = x^2 + 6x + 1 \)?
2. How does changing the coefficient of \( x^2 \) in a quadratic function affect the range?
3. What is the range of the function \( f(x) = -x^2 + 6x + 1 \)?
4. How can the vertex form of a quadratic function help in determining the range?
5. What transformations can be applied to the function \( f(x) = x^2 \) to obtain \( f(x) = x^2 + 6x + 1 \)?

**Tip:** The vertex of a parabola gives you the minimum or maximum value of the quadratic function, which is key to determining the range.

Discover how to find the range of the quadratic function f(x) = x^2 + 6x + 1 using graphing technology. Explore the vertex form and graph analysis with a step-by-step solution. Suitable for high school students studying quadratic functions and graphing.

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