Write $g(x) = 40x + 4x^2$ In Vertex Form.1. Write The Function In Standard Form: $g(x) = 4x^2 + 40x$2. Factor $a$ Out Of The First Two Terms: $g(x) = 4(x^2 + 10x$\]3. Complete The Square By Adding And Subtracting

Introduction

In mathematics, quadratic functions are a fundamental concept in algebra and calculus. They are used to model various real-world phenomena, such as the trajectory of a projectile, the growth of a population, and the motion of an object under the influence of a force. One of the most important forms of a quadratic function is the vertex form, which provides valuable information about the function's behavior and properties. In this article, we will learn how to convert a quadratic function from standard form to vertex form.

Standard Form to Vertex Form

The standard form of a quadratic function is given by:

$$g(x) = ax^2 + bx + c$$

where $a$, $b$, and $c$ are constants. To convert this function to vertex form, we need to complete the square. The vertex form of a quadratic function is given by:

$$g(x) = a(x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola.

Step 1: Write the Function in Standard Form

The given function is:

$$g(x) = 40x + 4x^2$$

We can rewrite this function in standard form by rearranging the terms:

$$g(x) = 4x^2 + 40x$$

Step 2: Factor $a$ out of the First Two Terms

Now, we can factor $a$ out of the first two terms:

$$g(x) = 4(x^2 + 10x)$$

Step 3: Complete the Square

To complete the square, we need to add and subtract $(b/2)^2$ inside the parentheses:

$$g(x) = 4(x^2 + 10x + 25 - 25)$$

Now, we can rewrite the expression inside the parentheses as a perfect square:

$$g(x) = 4((x + 5)^2 - 25)$$

Step 4: Simplify the Expression

We can simplify the expression by distributing the $4$:

$$g(x) = 4(x + 5)^2 - 100$$

Step 5: Write the Function in Vertex Form

Now, we can write the function in vertex form:

$$g(x) = 4(x + 5)^2 - 100$$

The vertex of the parabola is $(-5, -100)$.

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 function in vertex form. The vertex form of a quadratic function provides valuable information about the function's behavior and properties, such as the vertex, axis of symmetry, and direction of opening. We hope that this article has provided a clear and concise explanation of how to convert a quadratic function to vertex form.

Example Problems

1. Convert the function $g(x) = 3x^2 + 12x$ to vertex form.
2. Convert the function $g(x) = 2x^2 - 8x$ to vertex form.
3. Convert the function $g(x) = x^2 + 6x$ to vertex form.

Solutions

1. $g(x) = 3(x + 2)^2 - 12$
2. $g(x) = 2(x - 2)^2 - 8$
3. $g(x) = (x + 3)^2 - 9$

Discussion

The vertex form of a quadratic function is a powerful tool for analyzing and understanding the behavior of quadratic functions. It provides valuable information about the function's vertex, axis of symmetry, and direction of opening. By converting a quadratic function to vertex form, we can gain a deeper understanding of the function's properties and behavior.

References

1. "Algebra and Trigonometry" by Michael Sullivan
2. "Calculus" by James Stewart
3. "Mathematics for Computer Science" by Eric Lehman

Glossary

• Vertex form: A form of a quadratic function that provides valuable information about the function's behavior and properties.
• Completing the square: A method for rewriting a quadratic function in vertex form.
• Axis of symmetry: A line that passes through the vertex of a parabola and is perpendicular to the direction of opening.
• Direction of opening: The direction in which a parabola opens, either upward or downward.
  Vertex Form of a Quadratic Function: Q&A =============================================

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 the vertex form of a quadratic function.

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

A: The vertex form of a quadratic function is a form that provides valuable information about the function's behavior and properties. It is given by:

$$g(x) = a(x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola.

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

A: To convert a quadratic function to vertex form, you need to complete the square. This involves adding and subtracting $(b/2)^2$ inside the parentheses.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is also the minimum or maximum point of the parabola, depending on whether the parabola opens upward or downward.

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

A: To find the vertex of a parabola, you need to complete the square and rewrite the function in vertex form. The vertex is given by the point $(h, k)$ in the vertex form.

Q: What is the axis of symmetry of a parabola?

A: The axis of symmetry of a parabola is a line that passes through the vertex of the parabola and is perpendicular to the direction of opening.

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

A: To find the axis of symmetry of a parabola, you need to find the vertex of the parabola and then draw a line that passes through the vertex and is perpendicular to the direction of opening.

Q: What is the direction of opening of a parabola?

A: The direction of opening of a parabola is the direction in which the parabola opens, either upward or downward.

Q: How do I find the direction of opening of a parabola?

A: To find the direction of opening of a parabola, you need to look at the coefficient of the $x^2$ term. If the coefficient is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

Q: Can I use the vertex form of a quadratic function to graph the function?

A: Yes, you can use the vertex form of a quadratic function to graph the function. The vertex form provides valuable information about the function's behavior and properties, which can be used to graph the function.

Q: Are there any other forms of a quadratic function besides standard form and vertex form?

A: Yes, there are other forms of a quadratic function besides standard form and vertex form. Some other forms include:

• Factored form: A form that expresses the quadratic function as a product of two binomials.
• Completing the square form: A form that expresses the quadratic function as a perfect square trinomial.
• Graphical form: A form that expresses the quadratic function as a graph.

Conclusion

In this article, we answered some frequently asked questions about the vertex form of a quadratic function. We hope that this article has provided a clear and concise explanation of the vertex form and its significance.

Example Problems

1. Convert the function $g(x) = 3x^2 + 12x$ to vertex form.
2. Find the vertex of the parabola given by the function $g(x) = 2x^2 - 8x$.
3. Find the axis of symmetry of the parabola given by the function $g(x) = x^2 + 6x$.

Solutions

1. $g(x) = 3(x + 2)^2 - 12$
2. $(2, -8)$
3. $x = -3$

Discussion

The vertex form of a quadratic function is a powerful tool for analyzing and understanding the behavior of quadratic functions. It provides valuable information about the function's vertex, axis of symmetry, and direction of opening. By converting a quadratic function to vertex form, we can gain a deeper understanding of the function's properties and behavior.

References

1. "Algebra and Trigonometry" by Michael Sullivan
2. "Calculus" by James Stewart
3. "Mathematics for Computer Science" by Eric Lehman

Glossary

• Vertex form: A form of a quadratic function that provides valuable information about the function's behavior and properties.
• Completing the square: A method for rewriting a quadratic function in vertex form.
• Axis of symmetry: A line that passes through the vertex of a parabola and is perpendicular to the direction of opening.
• Direction of opening: The direction in which a parabola opens, either upward or downward.