The Vertex of the Quadratic Function

Oct 13, 2025 by windailys.com 7 views

Understanding the Key Aspects of the Quadratic Function $f(x) = -x^2 - 2x - 1$

Quadratic functions are a fundamental concept in mathematics, and understanding their key aspects is crucial for solving various problems in algebra, calculus, and other branches of mathematics. In this article, we will focus on the key aspects of the quadratic function $f(x) = -x^2 - 2x - 1$ . We will use the drop-down menus to describe the key aspects of this function.

The Vertex of the Quadratic Function

1. The vertex is the $\square$

The vertex of a quadratic function is the point at which the function changes from decreasing to increasing or vice versa. To find the vertex of the function $f(x) = -x^2 - 2x - 1$ , we can use the formula:

x = -\frac{b}{2a}

where $a$ and $b$ are the coefficients of the quadratic function. In this case, $a = -1$ and $b = -2$ . Plugging these values into the formula, we get:

x = -\frac{-2}{2(-1)} = -\frac{-2}{-2} = -1

Now that we have found the x-coordinate of the vertex, we can find the y-coordinate by plugging $x = -1$ into the function:

f(-1) = -(-1)^2 - 2(-1) - 1 = -1 + 2 - 1 = 0

Therefore, the vertex of the function $f(x) = -x^2 - 2x - 1$ is the point $(-1, 0)$ .

The Increasing and Decreasing Intervals of the Quadratic Function

2. The function is increasing $\square$

To determine the intervals on which the function is increasing or decreasing, we need to examine the sign of the derivative of the function. The derivative of the function $f(x) = -x^2 - 2x - 1$ is:

f'(x) = -2x - 2

We can find the critical points of the function by setting the derivative equal to zero and solving for $x$ :

-2x - 2 = 0 \Rightarrow x = -1

Now that we have found the critical point, we can examine the sign of the derivative on either side of the critical point. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.

To the left of the critical point $x = -1$ , the derivative is negative, so the function is decreasing. To the right of the critical point $x = -1$ , the derivative is positive, so the function is increasing.

Therefore, the function $f(x) = -x^2 - 2x - 1$ is decreasing on the interval $(-\infty, -1)$ and increasing on the interval $(-1, \infty)$ .

3. The function is decreasing $\square$

As we discussed earlier, the function $f(x) = -x^2 - 2x - 1$ is decreasing on the interval $(-\infty, -1)$ .

4. The function is increasing $\square$

As we discussed earlier, the function $f(x) = -x^2 - 2x - 1$ is increasing on the interval $(-1, \infty)$ .

The Domain of the Quadratic Function

5. The domain of the function is $\square$

The domain of a function is the set of all possible input values for which the function is defined. In the case of the quadratic function $f(x) = -x^2 - 2x - 1$ , the domain is all real numbers, or in interval notation:

(-\infty, \infty)

This is because the quadratic function is defined for all real values of $x$ .

In our previous article, we discussed the key aspects of the quadratic function $f(x) = -x^2 - 2x - 1$ . In this article, we will answer some frequently asked questions about this function.

Q: What is the vertex of the quadratic function $f(x) = -x^2 - 2x - 1$ ?

A: The vertex of the quadratic function $f(x) = -x^2 - 2x - 1$ is the point $(-1, 0)$ .

Q: Is the function $f(x) = -x^2 - 2x - 1$ increasing or decreasing?

A: The function $f(x) = -x^2 - 2x - 1$ is decreasing on the interval $(-\infty, -1)$ and increasing on the interval $(-1, \infty)$ .

Q: What is the domain of the quadratic function $f(x) = -x^2 - 2x - 1$ ?

A: The domain of the quadratic function $f(x) = -x^2 - 2x - 1$ is all real numbers, or in interval notation:

(-\infty, \infty)

Q: How do I find the x-intercepts of the quadratic function $f(x) = -x^2 - 2x - 1$ ?

A: To find the x-intercepts of the quadratic function $f(x) = -x^2 - 2x - 1$ , you can set the function equal to zero and solve for $x$ :

-x^2 - 2x - 1 = 0

Using the quadratic formula, we get:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where $a = -1$ , $b = -2$ , and $c = -1$ . Plugging these values into the formula, we get:

x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-1)(-1)}}{2(-1)}

Simplifying, we get:

x = \frac{2 \pm \sqrt{4 - 4}}{-2}

x = \frac{2 \pm \sqrt{0}}{-2}

x = \frac{2}{-2}

x = -1

Therefore, the x-intercept of the quadratic function $f(x) = -x^2 - 2x - 1$ is the point $(-1, 0)$ .

Q: How do I find the y-intercept of the quadratic function $f(x) = -x^2 - 2x - 1$ ?

A: To find the y-intercept of the quadratic function $f(x) = -x^2 - 2x - 1$ , you can plug $x = 0$ into the function:

f(0) = -(0)^2 - 2(0) - 1

f(0) = -1

Therefore, the y-intercept of the quadratic function $f(x) = -x^2 - 2x - 1$ is the point $(0, -1)$ .

Q: Can I graph the quadratic function $f(x) = -x^2 - 2x - 1$ ?

A: Yes, you can graph the quadratic function $f(x) = -x^2 - 2x - 1$ using a graphing calculator or a computer algebra system. The graph of the function is a parabola that opens downward.

In conclusion, we have answered some frequently asked questions about the quadratic function $f(x) = -x^2 - 2x - 1$ . We hope this article has been helpful in understanding the key aspects of this function.