The Following Equation Of A Quadratic Function Is Given In Standard Form:${ F(x) = 2x^2 + 8x + 11 } F I N D T H E S A M E E Q U A T I O N I N V E R T E X F O R M , T H E N E N T E R I T B E L O W . R O U N D Y O U R A N S W E R S T O T H E N E A R E S T T E N T H I F N E C E S S A R Y . Find The Same Equation In Vertex Form, Then Enter It Below. Round Your Answers To The Nearest Tenth If Necessary. F In D T H Es Am Ee Q U A T I O Nin V Er T E X F Or M , T H E N E N T Er I T B E L O W . R O U N D Yo U R An S W Ers T O T H E N E A Res Tt E N T Hi F N Ecess A Ry . [ F(x) = \square (x -

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Introduction

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The standard form of a quadratic function is given by the equation f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants. However, there is another form of a quadratic function called the vertex form, which is given by the equation f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola. In this article, we will learn how to convert a quadratic function from standard form to vertex form.

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 a way that allows us to easily identify the vertex of the parabola.

Step 1: Identify the Coefficients

The first step is to identify the coefficients aa, bb, and cc in the standard form of the quadratic function. In this case, we have a=2a = 2, b=8b = 8, and c=11c = 11.

Step 2: Write the Quadratic Function in the Form f(x)=a(x+b/2a)2+c−b2/4af(x) = a(x + b/2a)^2 + c - b^2/4a

Now, we need to write the quadratic function in the form f(x)=a(x+b/2a)2+c−b2/4af(x) = a(x + b/2a)^2 + c - b^2/4a. This involves adding and subtracting a constant term inside the parentheses.

import math

a = 2
b = 8
c = 11



constant_term = (b**2) / (4 * a)



vertex_x = -b / (2 * a)
vertex_y = c - constant_term



print(f"The vertex is ({vertex_x}, {vertex_y})")

Step 3: Simplify the Expression

Now that we have written the quadratic function in the form f(x)=a(x+b/2a)2+c−b2/4af(x) = a(x + b/2a)^2 + c - b^2/4a, we can simplify the expression by evaluating the constant term.

# Simplify the expression
simplified_expression = f"2(x + {b / (2 * a)})^2 + {c - (b**2) / (4 * a)}"


print(f"The simplified expression is {simplified_expression}")

Step 4: Write the Quadratic Function in Vertex Form

Now that we have simplified the expression, we can write the quadratic function in vertex form.

# Write the quadratic function in vertex form
vertex_form = f"f(x) = 2(x + {b / (2 * a)})^2 + {c - (b**2) / (4 * a)}"


print(f"The vertex form is {vertex_form}")

Conclusion

In this article, we learned how to convert a quadratic function from standard form to vertex form. We used the method of completing the square to rewrite the quadratic function in a way that allows us to easily identify the vertex of the parabola. We also used Python code to simplify the expression and write the quadratic function in vertex form.

The Final Answer

The final answer is f(x)=2(x+2)2+3\boxed{f(x) = 2(x + 2)^2 + 3}.

Discussion

  • What is the vertex form of a quadratic function?
  • How do you convert a quadratic function from standard form to vertex form?
  • What is the method of completing the square?
  • How do you use Python code to simplify the expression and write the quadratic function in vertex form?

References

  • [1] "Quadratic Functions" by Math Open Reference
  • [2] "Completing the Square" by Khan Academy
  • [3] "Vertex Form of a Quadratic Function" by Purplemath

Related Topics

  • Quadratic Functions
  • Completing the Square
  • Vertex Form of a Quadratic Function
  • Quadratic Equations

Keywords

  • Quadratic function
  • Standard form
  • Vertex form
  • Completing the square
  • Python code
  • Simplifying expressions
  • Vertex form of a quadratic function
    Quadratic Function Vertex Form: Frequently Asked Questions ===========================================================

Introduction

In our previous article, we learned how to convert a quadratic function from standard form to vertex form. In this article, we will answer some frequently asked questions about quadratic function vertex form.

Q&A

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is given by the equation 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: How do you 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 a way that allows you to easily identify the vertex of the parabola.

Q: What is the method of completing the square?

A: The method of completing the square is a technique used to rewrite a quadratic function in a way that allows you to easily identify the vertex of the parabola. It involves adding and subtracting a constant term inside the parentheses.

Q: How do you use Python code to simplify the expression and write the quadratic function in vertex form?

A: You can use Python code to simplify the expression and write the quadratic function in vertex form by using the following steps:

  1. Define the coefficients aa, bb, and cc in the standard form of the quadratic function.
  2. Calculate the constant term using the formula constant_term=(b2)/(4∗a)constant\_term = (b^2) / (4 * a).
  3. Calculate the vertex using the formula vertex_x=−b/(2∗a)vertex\_x = -b / (2 * a) and vertex_y=c−constant_termvertex\_y = c - constant\_term.
  4. Simplify the expression using the formula $simplified_expression = f"2(x + {b / (2 * a)})^2 + {c - (b**2) / (4 * a)}"`.
  5. Write the quadratic function in vertex form using the formula $vertex_form = f"f(x) = 2(x + {b / (2 * a)})^2 + {c - (b**2) / (4 * a)}"`.

Q: What is the final answer for the given quadratic function?

A: The final answer for the given quadratic function is f(x)=2(x+2)2+3\boxed{f(x) = 2(x + 2)^2 + 3}.

Q: What are some related topics to quadratic function vertex form?

A: Some related topics to quadratic function vertex form include:

  • Quadratic Functions
  • Completing the Square
  • Vertex Form of a Quadratic Function
  • Quadratic Equations

Q: What are some keywords related to quadratic function vertex form?

A: Some keywords related to quadratic function vertex form include:

  • Quadratic function
  • Standard form
  • Vertex form
  • Completing the square
  • Python code
  • Simplifying expressions
  • Vertex form of a quadratic function

Conclusion

In this article, we answered some frequently asked questions about quadratic function vertex form. We covered topics such as the vertex form of a quadratic function, converting a quadratic function from standard form to vertex form, the method of completing the square, and using Python code to simplify the expression and write the quadratic function in vertex form.

Discussion

  • What are some common mistakes to avoid when converting a quadratic function from standard form to vertex form?
  • How do you use the method of completing the square to rewrite a quadratic function in vertex form?
  • What are some real-world applications of quadratic function vertex form?
  • How do you use Python code to simplify the expression and write the quadratic function in vertex form?

References

  • [1] "Quadratic Functions" by Math Open Reference
  • [2] "Completing the Square" by Khan Academy
  • [3] "Vertex Form of a Quadratic Function" by Purplemath

Related Topics

  • Quadratic Functions
  • Completing the Square
  • Vertex Form of a Quadratic Function
  • Quadratic Equations

Keywords

  • Quadratic function
  • Standard form
  • Vertex form
  • Completing the square
  • Python code
  • Simplifying expressions
  • Vertex form of a quadratic function