The Vertex Form of the Function

Oct 13, 2025 by magicalceylontours.com 29 views

Understanding the Function h(x) = x^2 + 20x - 17

The given function h(x) = x^2 + 20x - 17 is a quadratic function, which can be analyzed to determine its key characteristics. In this article, we will explore the properties of the function, including its vertex, vertex form, and maximum value.

The Vertex of h(x)

The vertex of a quadratic function is the point at which the function changes direction, either from increasing to decreasing or vice versa. To find the vertex of the function h(x) = x^2 + 20x - 17, we can use the formula for the x-coordinate of the vertex, which is given by:

x = -b / 2a

In this case, a = 1 and b = 20. Plugging these values into the formula, we get:

x = -20 / (2 * 1) x = -20 / 2 x = -10

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging x = -10 into the function:

h(-10) = (-10)^2 + 20(-10) - 17 h(-10) = 100 - 200 - 17 h(-10) = -117

Therefore, the vertex of the function h(x) is indeed (-10, -117).

The Vertex Form of the Function

The vertex form of a quadratic function is given by:

h(x) = (x - h)^2 + k

where (h, k) is the vertex of the function. We have already found that the vertex of the function h(x) is (-10, -117). Therefore, we can write the vertex form of the function as:

h(x) = (x + 10)^2 - 117

This is indeed the vertex form of the function, as it matches the vertex we found earlier.

The Maximum Value of the Function

Since the vertex of the function is a minimum point, the function has a maximum value at the endpoints of its domain. However, the given function h(x) = x^2 + 20x - 17 is a quadratic function, which means it has a parabolic shape and does not have a maximum value in the classical sense. The function will continue to increase without bound as x approaches positive infinity or negative infinity.

However, if we are looking for the maximum value of the function within a specific interval, we can use the fact that the function is a quadratic function to find the maximum value. The maximum value of a quadratic function occurs at the vertex, which we have already found to be (-10, -117). Therefore, the maximum value of the function within the interval [-10, 10] is indeed -117.

Conclusion

In conclusion, the vertex of the function h(x) = x^2 + 20x - 17 is indeed (-10, -117), and the vertex form of the function is h(x) = (x + 10)^2 - 117. However, the function does not have a maximum value in the classical sense, as it will continue to increase without bound as x approaches positive infinity or negative infinity.
Frequently Asked Questions about the Function h(x) = x^2 + 20x - 17

In this article, we will address some of the most common questions about the function h(x) = x^2 + 20x - 17.

Q: What is the vertex of the function h(x) = x^2 + 20x - 17?

A: The vertex of the function h(x) = x^2 + 20x - 17 is (-10, -117).

Q: How do I find the vertex of the function h(x) = x^2 + 20x - 17?

A: To find the vertex of the function h(x) = x^2 + 20x - 17, you can use the formula for the x-coordinate of the vertex, which is given by:

x = -b / 2a

In this case, a = 1 and b = 20. Plugging these values into the formula, you get:

x = -20 / (2 * 1) x = -20 / 2 x = -10

Now that you have the x-coordinate of the vertex, you can find the y-coordinate by plugging x = -10 into the function:

h(-10) = (-10)^2 + 20(-10) - 17 h(-10) = 100 - 200 - 17 h(-10) = -117

Q: What is the vertex form of the function h(x) = x^2 + 20x - 17?

A: The vertex form of the function h(x) = x^2 + 20x - 17 is h(x) = (x + 10)^2 - 117.

Q: Does the function h(x) = x^2 + 20x - 17 have a maximum value?

A: The function h(x) = x^2 + 20x - 17 does not have a maximum value in the classical sense, as it will continue to increase without bound as x approaches positive infinity or negative infinity. However, if you are looking for the maximum value of the function within a specific interval, you can use the fact that the function is a quadratic function to find the maximum value. The maximum value of a quadratic function occurs at the vertex, which we have already found to be (-10, -117).

Q: How do I graph the function h(x) = x^2 + 20x - 17?

A: To graph the function h(x) = x^2 + 20x - 17, you can use a graphing calculator or a computer algebra system. You can also use the fact that the function is a quadratic function to graph it. The graph of a quadratic function is a parabola that opens upward or downward. In this case, the graph of the function h(x) = x^2 + 20x - 17 is a parabola that opens upward.

Q: Can I use the function h(x) = x^2 + 20x - 17 to model real-world phenomena?

A: Yes, you can use the function h(x) = x^2 + 20x - 17 to model real-world phenomena. For example, you can use this function to model the height of a projectile as a function of time, or to model the cost of a product as a function of the number of units produced.

Q: How do I find the roots of the function h(x) = x^2 + 20x - 17?

A: To find the roots of the function h(x) = x^2 + 20x - 17, you can use the quadratic formula, which is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = 20, and c = -17. Plugging these values into the formula, you get:

x = (-(20) ± √((20)^2 - 4(1)(-17))) / (2(1)) x = (-20 ± √(400 + 68)) / 2 x = (-20 ± √468) / 2 x = (-20 ± 21.6) / 2

Therefore, the roots of the function h(x) = x^2 + 20x - 17 are x = -20.8 and x = 0.8.

Conclusion

In this article, we have addressed some of the most common questions about the function h(x) = x^2 + 20x - 17. We have discussed the vertex of the function, the vertex form of the function, and the maximum value of the function. We have also provided answers to some of the most common questions about the function, including how to find the vertex of the function, how to graph the function, and how to find the roots of the function.