Write The Equation For The Function:$f(x) = -x^2 - 1$

Introduction

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is given by $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this article, we will focus on the equation of a quadratic function, specifically the function $f(x) = -x^2 - 1$.

The Equation of a Quadratic Function

A quadratic function can be represented by the equation $f(x) = ax^2 + bx + c$. The graph of a quadratic function is a parabola, which is a U-shaped curve. The parabola can open upwards or downwards, depending on the value of $a$. If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards.

The Function $f(x) = -x^2 - 1$

The function $f(x) = -x^2 - 1$ is a quadratic function with a negative leading coefficient. This means that the parabola opens downwards. The graph of this function is a downward-facing parabola that intersects the y-axis at the point $(0, -1)$.

Graphing the Function

To graph the function $f(x) = -x^2 - 1$, we can start by finding the x-intercepts of the function. The x-intercepts are the points where the graph of the function crosses the x-axis. To find the x-intercepts, we set $f(x) = 0$ and solve for $x$.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = -x**2 - 1

# Solve for x
x_intercepts = sp.solve(f, x)

print(x_intercepts)

The output of the code above is [], which means that the function $f(x) = -x^2 - 1$ does not have any real x-intercepts. This is because the graph of the function is a downward-facing parabola that does not intersect the x-axis.

Finding the Vertex of the Parabola

The vertex of a parabola is the point where the parabola changes direction. The vertex of a quadratic function can be found using the formula $x = -\frac{b}{2a}$. In this case, $a = -1$ and $b = 0$, so the x-coordinate of the vertex is $x = -\frac{0}{2(-1)} = 0$.

To find the y-coordinate of the vertex, we substitute $x = 0$ into the function $f(x) = -x^2 - 1$. This gives us $f(0) = -(0)^2 - 1 = -1$.

Therefore, the vertex of the parabola is the point $(0, -1)$.

Conclusion

In this article, we have discussed the equation of a quadratic function, specifically the function $f(x) = -x^2 - 1$. We have graphed the function and found the x-intercepts and vertex of the parabola. The graph of the function is a downward-facing parabola that intersects the y-axis at the point $(0, -1)$. The vertex of the parabola is the point $(0, -1)$.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Graphing Quadratic Functions" by Purplemath

Further Reading

• "Quadratic Equations" by Khan Academy
• "Graphing Quadratic Functions" by Mathway

Code

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = -x**2 - 1

# Solve for x
x_intercepts = sp.solve(f, x)

print(x_intercepts)

$$

Introduction

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

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is given by $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What is the graph of a quadratic function?

A: The graph of a quadratic function is a parabola, which is a U-shaped curve. The parabola can open upwards or downwards, depending on the value of $a$. If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can start by finding the x-intercepts of the function. The x-intercepts are the points where the graph of the function crosses the x-axis. To find the x-intercepts, you set $f(x) = 0$ and solve for $x$.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. The vertex of a quadratic function can be found using the formula $x = -\frac{b}{2a}$. To find the y-coordinate of the vertex, you substitute $x = -\frac{b}{2a}$ into the function $f(x) = ax^2 + bx + c$.

Q: How do I find the x-intercepts of a quadratic function?

A: To find the x-intercepts of a quadratic function, you set $f(x) = 0$ and solve for $x$. This can be done using algebraic methods or numerical methods.

Q: What is the difference between a quadratic function and a linear function?

A: A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. The general form of a linear function is given by $f(x) = mx + b$, where $m$ and $b$ are constants.

Q: Can a quadratic function have more than one x-intercept?

A: Yes, a quadratic function can have more than one x-intercept. This occurs when the graph of the function intersects the x-axis at more than one point.

Q: How do I determine the direction of a parabola?

A: To determine the direction of a parabola, you look at the value of $a$ in the equation $f(x) = ax^2 + bx + c$. If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards.

Q: Can a quadratic function have a negative leading coefficient?

A: Yes, a quadratic function can have a negative leading coefficient. This occurs when the value of $a$ in the equation $f(x) = ax^2 + bx + c$ is negative.

Q: How do I graph a quadratic function with a negative leading coefficient?

A: To graph a quadratic function with a negative leading coefficient, you can start by finding the x-intercepts of the function. The x-intercepts are the points where the graph of the function crosses the x-axis. To find the x-intercepts, you set $f(x) = 0$ and solve for $x$.

Conclusion

In this article, we have answered some frequently asked questions about quadratic functions. We have discussed the graph of a quadratic function, how to graph a quadratic function, and how to find the x-intercepts and vertex of a parabola. We have also discussed the difference between a quadratic function and a linear function, and how to determine the direction of a parabola.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Graphing Quadratic Functions" by Purplemath

Further Reading

• "Quadratic Equations" by Khan Academy
• "Graphing Quadratic Functions" by Mathway

Code

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = -x**2 - 1

# Solve for x
x_intercepts = sp.solve(f, x)

print(x_intercepts)

This code can be used to find the x-intercepts of the function $f(x) = -x^2 - 1$. The output of the code is [], which means that the function does not have any real x-intercepts.