$f(x) = X^2 + 2x - 1$1. What Is The Axis Of Symmetry? $\square$2. What Is The Vertex? ( □ , □ (\square, \square ( □ , □ ]3. Does This Function Have A Minimum Or Maximum? $\square$4. What Is The Y Y Y -intercept?

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 $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this article, we will focus on the quadratic function $f(x) = x^2 + 2x - 1$ and explore its axis of symmetry, vertex, minimum or maximum, and $y$-intercept.

Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. It is denoted by the equation $x = -\frac{b}{2a}$. To find the axis of symmetry of the given quadratic function, we need to identify the values of $a$ and $b$.

import sympy as sp
x = sp.symbols('x')

f = x**2 + 2*x - 1

a = f.coeff(x, 2)
b = f.coeff(x, 1)

axis_of_symmetry = -b / (2 * a)
print(axis_of_symmetry)

The axis of symmetry is $x = -\frac{2}{2(1)} = -1$.

Vertex

The vertex of a quadratic function is the point on the parabola where it reaches its maximum or minimum value. It is denoted by the coordinates $(h, k)$, where $h$ is the $x$-coordinate of the vertex and $k$ is the $y$-coordinate of the vertex. To find the vertex of the given quadratic function, we need to use the formula $h = -\frac{b}{2a}$ and $k = f(h)$.

# Calculate the x-coordinate of the vertex
h = -b / (2 * a)

k = f.subs(x, h)
print((h, k))

The vertex is $(-1, -2)$.

Minimum or Maximum

A quadratic function can have either a minimum or a maximum value, depending on the sign of the coefficient $a$. If $a > 0$, the function has a minimum value, and if $a < 0$, the function has a maximum value. In this case, $a = 1 > 0$, so the function has a minimum value.

$y$-Intercept

The $y$-intercept of a quadratic function is the point where the function intersects the $y$-axis. It is denoted by the coordinates $(0, c)$, where $c$ is the constant term in the function. To find the $y$-intercept of the given quadratic function, we need to substitute $x = 0$ into the function.

# Calculate the y-intercept
y_intercept = f.subs(x, 0)
print(y_intercept)

The $y$-intercept is $-1$.

Conclusion

In conclusion, the axis of symmetry of the quadratic function $f(x) = x^2 + 2x - 1$ is $x = -1$, the vertex is $(-1, -2)$, the function has a minimum value, and the $y$-intercept is $-1$. These values provide valuable information about the behavior of the function and can be used to graph the function and analyze its properties.

References

• [1] Khan Academy. (n.d.). Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/quadratic-functions
• [2] Wolfram MathWorld. (n.d.). Quadratic Function. Retrieved from https://mathworld.wolfram.com/QuadraticFunction.html

Discussion

• What is the significance of the axis of symmetry in a quadratic function?
• How does the vertex of a quadratic function relate to the axis of symmetry?
• What is the difference between a minimum and a maximum value in a quadratic function?
• How can the $y$-intercept of a quadratic function be used to analyze its behavior?
  Quadratic Function Q&A ==========================

Frequently Asked Questions

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 $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What is the axis of symmetry of a quadratic function?

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. It is denoted by the equation $x = -\frac{b}{2a}$.

Q: How do I find the axis of symmetry of a quadratic function?

A: To find the axis of symmetry of a quadratic function, you need to identify the values of $a$ and $b$ in the function. Then, you can use the formula $x = -\frac{b}{2a}$ to calculate the axis of symmetry.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the parabola where it reaches its maximum or minimum value. It is denoted by the coordinates $(h, k)$, where $h$ is the $x$-coordinate of the vertex and $k$ is the $y$-coordinate of the vertex.

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you need to use the formula $h = -\frac{b}{2a}$ to calculate the $x$-coordinate of the vertex. Then, you can substitute $x = h$ into the function to calculate the $y$-coordinate of the vertex.

Q: Does a quadratic function have a minimum or maximum value?

A: A quadratic function can have either a minimum or a maximum value, depending on the sign of the coefficient $a$. If $a > 0$, the function has a minimum value, and if $a < 0$, the function has a maximum value.

Q: How do I determine whether a quadratic function has a minimum or maximum value?

A: To determine whether a quadratic function has a minimum or maximum value, you need to examine the sign of the coefficient $a$. If $a > 0$, the function has a minimum value, and if $a < 0$, the function has a maximum value.

Q: What is the $y$-intercept of a quadratic function?

A: The $y$-intercept of a quadratic function is the point where the function intersects the $y$-axis. It is denoted by the coordinates $(0, c)$, where $c$ is the constant term in the function.

Q: How do I find the $y$-intercept of a quadratic function?

A: To find the $y$-intercept of a quadratic function, you need to substitute $x = 0$ into the function.

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

A: No, a quadratic function can have only one $y$-intercept.

Q: Can a quadratic function have more than one axis of symmetry?

A: No, a quadratic function can have only one axis of symmetry.

Q: Can a quadratic function have more than one vertex?

A: No, a quadratic function can have only one vertex.

Additional Resources

• [1] Khan Academy. (n.d.). Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/quadratic-functions
• [2] Wolfram MathWorld. (n.d.). Quadratic Function. Retrieved from https://mathworld.wolfram.com/QuadraticFunction.html

Discussion

• What is the most important concept to understand when working with quadratic functions?
• How do you determine whether a quadratic function has a minimum or maximum value?
• What is the significance of the axis of symmetry in a quadratic function?
• How can the $y$-intercept of a quadratic function be used to analyze its behavior?