Math Problem Statement

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+6x+1.

Solution

To find the range of the function f(x)=x2+6x+1f(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)=x2+6x+1f(x) = x^2 + 6x + 1 is a quadratic function in standard form ax2+bx+cax^2 + bx + c, where a=1a = 1, b=6b = 6, and c=1c = 1.

Vertex of the Parabola

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

xvertex=b2a=62(1)=3x_{\text{vertex}} = -\frac{b}{2a} = -\frac{6}{2(1)} = -3

To find the corresponding yy value at this xx:

f(3)=(3)2+6(3)+1=918+1=8f(-3) = (-3)^2 + 6(-3) + 1 = 9 - 18 + 1 = -8

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

Graphing the Function

The function f(x)=x2+6x+1f(x) = x^2 + 6x + 1 is a parabola that opens upwards (since the coefficient of x2x^2 is positive). The vertex is at (3,8)(-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=8y = -8, the range of f(x)f(x) is all values of yy that are greater than or equal to 8-8.

Conclusion

Thus, the range of the function f(x)=x2+6x+1f(x) = x^2 + 6x + 1 is: Range=[8,)\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)=x2+6x+1f(x) = x^2 + 6x + 1?
  2. How does changing the coefficient of x2x^2 in a quadratic function affect the range?
  3. What is the range of the function f(x)=x2+6x+1f(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)=x2f(x) = x^2 to obtain f(x)=x2+6x+1f(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.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Vertex Form
Graphing

Formulas

Vertex formula for a quadratic function

Theorems

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Suitable Grade Level

Grades 9-12