Which Function In Vertex Form Is Equivalent To $f(x)=x^2+6x+3$?A. $f(x)=(x+3)^2+3$B. $f(x)=(x+3)^2-6$C. $f(x)=(x+6)^2+3$D. $f(x)=(x+6)^2-6$

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Introduction

In mathematics, the vertex form of a quadratic function is a powerful tool for representing and analyzing quadratic equations. It provides a convenient way to express quadratic functions in a form that highlights their vertex, or turning point. In this article, we will explore the concept of vertex form and learn how to convert a quadratic function from standard form to vertex form.

What is Vertex Form?

Vertex form is a way of expressing a quadratic function in the form f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola. The vertex form is particularly useful for identifying the vertex of a parabola, as well as for graphing and analyzing quadratic functions.

Converting from Standard Form to Vertex Form

To convert a quadratic function from standard form to vertex form, we need to complete the square. This involves rewriting the quadratic function in the form f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where hh and kk are constants.

Step 1: Identify the Coefficients

The first step in converting a quadratic function from standard form to vertex form is to identify the coefficients of the quadratic function. The standard form of a quadratic function is f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants.

Step 2: Complete the Square

Once we have identified the coefficients, we can complete the square by rewriting the quadratic function in the form f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k. To do this, we need to find the value of hh that makes the expression (x−h)2(x - h)^2 equal to the original quadratic function.

Step 3: Simplify the Expression

Once we have completed the square, we can simplify the expression by combining like terms.

Example: Converting f(x)=x2+6x+3f(x) = x^2 + 6x + 3 to Vertex Form

Let's use the quadratic function f(x)=x2+6x+3f(x) = x^2 + 6x + 3 as an example. To convert this function to vertex form, we need to complete the square.

Step 1: Identify the Coefficients

The coefficients of the quadratic function are a=1a = 1, b=6b = 6, and c=3c = 3.

Step 2: Complete the Square

To complete the square, we need to find the value of hh that makes the expression (x−h)2(x - h)^2 equal to the original quadratic function. We can do this by using the formula h=−b2ah = -\frac{b}{2a}.

import math

# Define the coefficients
a = 1
b = 6
c = 3

# Calculate the value of h
h = -b / (2 * a)
print(h)

Step 3: Simplify the Expression

Once we have completed the square, we can simplify the expression by combining like terms.

# Define the value of h
h = -3

# Define the expression
expression = (x - h)**2 + c

# Simplify the expression
simplified_expression = (x + 3)**2 + 3
print(simplified_expression)

Conclusion

In this article, we have learned how to convert a quadratic function from standard form to vertex form. We have used the quadratic function f(x)=x2+6x+3f(x) = x^2 + 6x + 3 as an example and have shown how to complete the square and simplify the expression. We have also discussed the importance of vertex form in mathematics and have provided a step-by-step guide on how to convert a quadratic function from standard form to vertex form.

Which Function in Vertex Form is Equivalent to f(x)=x2+6x+3f(x) = x^2 + 6x + 3?

Based on our calculations, we can see that the function in vertex form that is equivalent to f(x)=x2+6x+3f(x) = x^2 + 6x + 3 is:

f(x)=(x+3)2+3f(x) = (x + 3)^2 + 3

This is option A in the given choices.

Q&A: Vertex Form and Quadratic Functions

Q: What is vertex form?

A: Vertex form is a way of expressing a quadratic function in the form f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.

Q: Why is vertex form important?

A: Vertex form is important because it provides a convenient way to express quadratic functions in a form that highlights their vertex, or turning point. This is particularly useful for identifying the vertex of a parabola, as well as for graphing and analyzing quadratic functions.

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

A: To convert a quadratic function from standard form to vertex form, you need to complete the square. This involves rewriting the quadratic function in the form f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where hh and kk are constants.

Q: What are the steps to complete the square?

A: The steps to complete the square are:

  1. Identify the coefficients of the quadratic function.
  2. Complete the square by rewriting the quadratic function in the form f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k.
  3. Simplify the expression by combining like terms.

Q: How do I find the value of h?

A: To find the value of hh, you can use the formula h=−b2ah = -\frac{b}{2a}.

Q: What is the significance of the vertex in a quadratic function?

A: The vertex of a quadratic function is the point at which the parabola changes direction. It is also the minimum or maximum point of the parabola, depending on the direction of the parabola.

Q: How do I graph a quadratic function in vertex form?

A: To graph a quadratic function in vertex form, you can use the following steps:

  1. Identify the vertex of the parabola.
  2. Determine the direction of the parabola.
  3. Plot the vertex on a coordinate plane.
  4. Plot additional points on the parabola using the equation f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k.

Q: What are some common mistakes to avoid when converting a quadratic function from standard form to vertex form?

A: Some common mistakes to avoid when converting a quadratic function from standard form to vertex form include:

  • Failing to identify the coefficients of the quadratic function.
  • Failing to complete the square correctly.
  • Failing to simplify the expression by combining like terms.

Q: How do I check my work when converting a quadratic function from standard form to vertex form?

A: To check your work when converting a quadratic function from standard form to vertex form, you can use the following steps:

  1. Verify that the vertex form is in the correct form.
  2. Verify that the coefficients of the quadratic function are correct.
  3. Verify that the expression is simplified by combining like terms.

Conclusion

In this article, we have provided a comprehensive guide to converting quadratic functions from standard form to vertex form. We have also answered some common questions about vertex form and quadratic functions. By following the steps outlined in this article, you should be able to convert a quadratic function from standard form to vertex form with ease.

Which Function in Vertex Form is Equivalent to f(x)=x2+6x+3f(x) = x^2 + 6x + 3?

Based on our calculations, we can see that the function in vertex form that is equivalent to f(x)=x2+6x+3f(x) = x^2 + 6x + 3 is:

f(x)=(x+3)2+3f(x) = (x + 3)^2 + 3

This is option A in the given choices.

Answer: A