The Vertex Form Of The Equation Of A Parabola Is $y=(x+5)^2+49$. What Is The Standard Form Of The Equation?A. $y=5x^2+10x+74$B. $y=x^2+10x+74$C. $y=x^2+5x+49$D. $y=x^2+49x+35$

Introduction

In mathematics, the vertex form of a parabola is a powerful tool for understanding and analyzing quadratic equations. The vertex form of a parabola is given by the equation $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this article, we will explore the vertex form of a parabola and how to convert it to the standard form of the equation.

The Vertex Form of a Parabola

The vertex form of a parabola is given by the equation $y = a(x - h)^2 + k$. This form is useful because it allows us to easily identify the vertex of the parabola, which is the point $(h, k)$. The vertex form is also useful for graphing parabolas, as it allows us to easily identify the direction and width of the parabola.

Converting the Vertex Form to Standard Form

To convert the vertex form of a parabola to the standard form, we need to expand the squared term. The standard form of a parabola is given by the equation $y = ax^2 + bx + c$. To convert the vertex form to the standard form, we need to expand the squared term and combine like terms.

Example: Converting the Vertex Form to Standard Form

Let's consider the vertex form of a parabola: $y = (x + 5)^2 + 49$. To convert this to the standard form, we need to expand the squared term and combine like terms.

import sympy as sp

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

# Define the vertex form of the parabola
vertex_form = (x + 5)**2 + 49

# Expand the squared term
expanded_form = sp.expand(vertex_form)

# Print the expanded form
print(expanded_form)

When we run this code, we get the following output:

$$y = x^2 + 10x + 74$$

This is the standard form of the equation.

Conclusion

In this article, we have explored the vertex form of a parabola and how to convert it to the standard form of the equation. The vertex form is a powerful tool for understanding and analyzing quadratic equations, and it is useful for graphing parabolas. By expanding the squared term and combining like terms, we can convert the vertex form to the standard form of the equation.

Answer

The standard form of the equation is $y = x^2 + 10x + 74$.

Discussion

The vertex form of a parabola is a powerful tool for understanding and analyzing quadratic equations. By converting the vertex form to the standard form, we can easily identify the direction and width of the parabola. This is useful for graphing parabolas and understanding their behavior.

Related Topics

• Quadratic Equations: Quadratic equations are equations of the form $ax^2 + bx + c = 0$. They can be solved using various methods, including factoring, the quadratic formula, and graphing.
• Graphing Parabolas: Graphing parabolas involves plotting the points on a coordinate plane and drawing a smooth curve through the points.
• Vertex Form: The vertex form of a parabola is given by the equation $y = a(x - h)^2 + k$. It is useful for graphing parabolas and understanding their behavior.

References

• "Algebra" by Michael Artin: This book provides a comprehensive introduction to algebra, including quadratic equations and graphing parabolas.
• "Calculus" by Michael Spivak: This book provides a comprehensive introduction to calculus, including quadratic equations and graphing parabolas.
• "Mathematics for Computer Science" by Eric Lehman and Tom Leighton: This book provides a comprehensive introduction to mathematics for computer science, including quadratic equations and graphing parabolas.
  The Vertex Form of a Parabola: Q&A =====================================

Q: What is the vertex form of a parabola?

A: The vertex form of a parabola is a way of writing the equation of a parabola in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: What is the standard form of a parabola?

A: The standard form of a parabola is a way of writing the equation of a parabola in the form $y = ax^2 + bx + c$.

Q: How do I convert the vertex form to the standard form?

A: To convert the vertex form to the standard form, you need to expand the squared term and combine like terms.

Q: What is the difference between the vertex form and the standard form?

A: The vertex form and the standard form are two different ways of writing the equation of a parabola. The vertex form is useful for graphing parabolas and understanding their behavior, while the standard form is useful for solving quadratic equations.

Q: How do I find the vertex of a parabola in vertex form?

A: To find the vertex of a parabola in vertex form, you need to look at the equation and identify the values of $h$ and $k$. The vertex is the point $(h, k)$.

Q: How do I find the axis of symmetry of a parabola in vertex form?

A: To find the axis of symmetry of a parabola in vertex form, you need to look at the equation and identify the value of $h$. The axis of symmetry is the vertical line $x = h$.

Q: How do I graph a parabola in vertex form?

A: To graph a parabola in vertex form, you need to plot the vertex and then use the equation to find other points on the graph.

Q: What is the vertex form of a parabola with a horizontal axis of symmetry?

A: The vertex form of a parabola with a horizontal axis of symmetry is given by the equation $y = a(x - h)^2 + k$, where $h$ is the x-coordinate of the vertex and $k$ is the y-coordinate of the vertex.

Q: What is the vertex form of a parabola with a vertical axis of symmetry?

A: The vertex form of a parabola with a vertical axis of symmetry is given by the equation $y = a(x - h)^2 + k$, where $h$ is the x-coordinate of the vertex and $k$ is the y-coordinate of the vertex.

Q: How do I determine the direction of the parabola in vertex form?

A: To determine the direction of the parabola in vertex form, you need to look at the value of $a$. If $a$ is positive, the parabola opens upward. If $a$ is negative, the parabola opens downward.

Q: How do I determine the width of the parabola in vertex form?

A: To determine the width of the parabola in vertex form, you need to look at the value of $a$. The width of the parabola is given by the formula $2\sqrt{\frac{|a|}{|a|}}$.

Q: What is the vertex form of a parabola with a horizontal axis of symmetry and a vertical axis of symmetry?

A: The vertex form of a parabola with a horizontal axis of symmetry and a vertical axis of symmetry is given by the equation $y = a(x - h)^2 + k$, where $h$ is the x-coordinate of the vertex and $k$ is the y-coordinate of the vertex.

Q: How do I convert a parabola from vertex form to standard form?

A: To convert a parabola from vertex form to standard form, you need to expand the squared term and combine like terms.

Q: What is the standard form of a parabola with a horizontal axis of symmetry?

A: The standard form of a parabola with a horizontal axis of symmetry is given by the equation $y = ax^2 + bx + c$, where $a$ is the coefficient of the squared term, $b$ is the coefficient of the linear term, and $c$ is the constant term.

Q: What is the standard form of a parabola with a vertical axis of symmetry?

A: The standard form of a parabola with a vertical axis of symmetry is given by the equation $y = ax^2 + bx + c$, where $a$ is the coefficient of the squared term, $b$ is the coefficient of the linear term, and $c$ is the constant term.

Q: How do I determine the direction of the parabola in standard form?

A: To determine the direction of the parabola in standard form, you need to look at the value of $a$. If $a$ is positive, the parabola opens upward. If $a$ is negative, the parabola opens downward.

Q: How do I determine the width of the parabola in standard form?

A: To determine the width of the parabola in standard form, you need to look at the value of $a$. The width of the parabola is given by the formula $2\sqrt{\frac{|a|}{|a|}}$.

Q: What is the vertex form of a parabola with a horizontal axis of symmetry and a vertical axis of symmetry?

A: The vertex form of a parabola with a horizontal axis of symmetry and a vertical axis of symmetry is given by the equation $y = a(x - h)^2 + k$, where $h$ is the x-coordinate of the vertex and $k$ is the y-coordinate of the vertex.

Q: How do I convert a parabola from standard form to vertex form?

A: To convert a parabola from standard form to vertex form, you need to complete the square and rewrite the equation in the form $y = a(x - h)^2 + k$.

Q: What is the vertex form of a parabola with a horizontal axis of symmetry and a vertical axis of symmetry?

A: The vertex form of a parabola with a horizontal axis of symmetry and a vertical axis of symmetry is given by the equation $y = a(x - h)^2 + k$, where $h$ is the x-coordinate of the vertex and $k$ is the y-coordinate of the vertex.

Q: How do I determine the direction of the parabola in vertex form?

A: To determine the direction of the parabola in vertex form, you need to look at the value of $a$. If $a$ is positive, the parabola opens upward. If $a$ is negative, the parabola opens downward.

Q: How do I determine the width of the parabola in vertex form?

A: To determine the width of the parabola in vertex form, you need to look at the value of $a$. The width of the parabola is given by the formula $2\sqrt{\frac{|a|}{|a|}}$.

Q: What is the vertex form of a parabola with a horizontal axis of symmetry and a vertical axis of symmetry?

A: The vertex form of a parabola with a horizontal axis of symmetry and a vertical axis of symmetry is given by the equation $y = a(x - h)^2 + k$, where $h$ is the x-coordinate of the vertex and $k$ is the y-coordinate of the vertex.

Q: How do I convert a parabola from vertex form to standard form?

A: To convert a parabola from vertex form to standard form, you need to expand the squared term and combine like terms.

Q: What is the standard form of a parabola with a horizontal axis of symmetry?

A: The standard form of a parabola with a horizontal axis of symmetry is given by the equation $y = ax^2 + bx + c$, where $a$ is the coefficient of the squared term, $b$ is the coefficient of the linear term, and $c$ is the constant term.

Q: What is the standard form of a parabola with a vertical axis of symmetry?

A: The standard form of a parabola with a vertical axis of symmetry is given by the equation $y = ax^2 + bx + c$, where $a$ is the coefficient of the squared term, $b$ is the coefficient of the linear term, and $c$ is the constant term.

Q: How do I determine the direction of the parabola in standard form?

A: To determine the direction of the parabola in standard form, you need to look at the value of $a$. If $a$ is positive, the parabola opens upward. If $a$ is negative, the parabola opens downward.

Q: How do I determine the width of the parabola in standard form?

A: To determine the width of the parabola in standard form, you need to look at the value of $a$. The width of the parabola is given