Understanding the Graph of a Quadratic Function

Oct 13, 2025 by miodeya.com 19 views

**The Graph of the Function $f(x)=-(x+1)^2$: A Comprehensive Guide**

Understanding the Graph of a Quadratic Function

The graph of a quadratic function is a parabola, which is a U-shaped curve. The graph of the function $f(x)=-(x+1)^2$ is a downward-facing parabola, meaning that it opens downwards. In this article, we will explore the key aspects of the graph of this function and provide a comprehensive guide to understanding its behavior.

Key Aspects of the Graph

1. The Vertex

The vertex of a parabola is the point at which the parabola changes direction. In the case of the function $f(x)=-(x+1)^2$ , the vertex is the point $(-1, 0)$ .

2. The Function is Positive

The function $f(x)=-(x+1)^2$ is a quadratic function with a negative leading coefficient. This means that the function is negative for all values of $x$ . However, the graph of the function is a downward-facing parabola, which means that the function is positive at the vertex.

3. The Function is Decreasing

The function $f(x)=-(x+1)^2$ is a decreasing function, meaning that as $x$ increases, the value of the function decreases.

Q&A

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

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

Q: Is the function $f(x)=-(x+1)^2$ positive or negative?

A: The function $f(x)=-(x+1)^2$ is negative for all values of $x$ . However, the graph of the function is a downward-facing parabola, which means that the function is positive at the vertex.

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

A: The function $f(x)=-(x+1)^2$ is a decreasing function, meaning that as $x$ increases, the value of the function decreases.

Q: What is the equation of the axis of symmetry of the graph of the function $f(x)=-(x+1)^2$ ?

A: The equation of the axis of symmetry of the graph of the function $f(x)=-(x+1)^2$ is $x=-1$ .

Q: What is the y-intercept of the graph of the function $f(x)=-(x+1)^2$ ?

A: The y-intercept of the graph of the function $f(x)=-(x+1)^2$ is the point $(0, -1)$ .

Q: What is the x-intercept of the graph of the function $f(x)=-(x+1)^2$ ?

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

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

A: The vertex form of the function $f(x)=-(x+1)^2$ is $f(x)=-[x-(-1)]^2$ .

Q: What is the standard form of the function $f(x)=-(x+1)^2$ ?

A: The standard form of the function $f(x)=-(x+1)^2$ is $f(x)=-x^2-2x-1$ .

Conclusion

In this article, we have explored the key aspects of the graph of the function $f(x)=-(x+1)^2$ . We have discussed the vertex, the function being positive or negative, and the function being increasing or decreasing. We have also provided a comprehensive guide to understanding the behavior of the graph of this function.