What Is $f(x)=7x^2+42x$ Written In Vertex Form?A. $f(x)=7(x+6)^2-6$ B. $f(x)=7(x+6)^2-42$ C. $f(x)=7(x+3)^2-9$ D. $f(x)=7(x+3)^2-63$

Understanding the Basics of Vertex Form

Vertex form is a way of expressing a quadratic function in the form of $f(x) = a(x - h)^2 + k$, where $(h, k)$ represents the vertex of the parabola. This form is particularly useful for identifying the vertex and the direction of the parabola's opening. In this article, we will explore how to convert a given quadratic function to vertex form.

The Given Quadratic Function

The given quadratic function is $f(x) = 7x^2 + 42x$. To convert this function to vertex form, we need to complete the square.

Completing the Square

To complete the square, we start by factoring out the coefficient of $x^2$, which is 7. This gives us:

$f(x) = 7(x^2 + 6x)$

Next, we add and subtract the square of half the coefficient of $x$ inside the parentheses. The coefficient of $x$ is 6, so half of it is 3, and its square is 9. We add and subtract 9 inside the parentheses:

$f(x) = 7(x^2 + 6x + 9 - 9)$

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

$f(x) = 7((x + 3)^2 - 9)$

Simplifying the Expression

We can simplify the expression by distributing the 7 to the terms inside the parentheses:

$f(x) = 7(x + 3)^2 - 63$

Comparing with the Options

Now that we have the quadratic function in vertex form, we can compare it with the given options:

A. $f(x) = 7(x + 6)^2 - 6$ B. $f(x) = 7(x + 6)^2 - 42$ C. $f(x) = 7(x + 3)^2 - 9$ D. $f(x) = 7(x + 3)^2 - 63$

Our derived vertex form matches option C.

Conclusion

In this article, we have learned how to convert a quadratic function to vertex form by completing the square. We started with the given quadratic function $f(x) = 7x^2 + 42x$ and factored out the coefficient of $x^2$, then added and subtracted the square of half the coefficient of $x$ inside the parentheses. We simplified the expression and compared it with the given options. The correct answer is option C, $f(x) = 7(x + 3)^2 - 9$.

Key Takeaways

• To convert a quadratic function to vertex form, we need to complete the square.
• We factor out the coefficient of $x^2$ and add and subtract the square of half the coefficient of $x$ inside the parentheses.
• We simplify the expression and compare it with the given options.

Practice Problems

1. Convert the quadratic function $f(x) = 2x^2 + 12x$ to vertex form.
2. Convert the quadratic function $f(x) = 5x^2 - 20x$ to vertex form.
3. Convert the quadratic function $f(x) = 3x^2 + 18x$ to vertex form.

Solutions

1. $f(x) = 2(x + 6)^2 - 36$
2. $f(x) = 5(x - 4)^2 - 20$
3. $f(x) = 3(x + 6)^2 - 54$

Additional Resources

For more practice problems and solutions, check out the following resources:

• Khan Academy: Quadratic Functions
• Mathway: Quadratic Functions
• Purplemath: Quadratic Functions

Final Thoughts

Frequently Asked Questions

In this article, we will address some of the most common questions related to quadratic functions, including their vertex form, graphing, and applications.

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

A: The vertex form of a quadratic function is $f(x) = a(x - h)^2 + k$, where $(h, k)$ represents 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 factoring out the coefficient of $x^2$, adding and subtracting the square of half the coefficient of $x$ inside the parentheses, and simplifying the expression.

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

A: The vertex of a quadratic function represents the maximum or minimum point of the parabola. In the vertex form, the vertex is represented by the point $(h, k)$.

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 $(h, k)$.
2. Determine the direction of the parabola's opening (upward or downward).
3. Plot the vertex on the graph.
4. Use the vertex as a reference point to plot additional points on the graph.

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have numerous real-world applications, including:

1. Projectile motion: Quadratic functions can be used to model the trajectory of a projectile under the influence of gravity.
2. Optimization problems: Quadratic functions can be used to solve optimization problems, such as finding the maximum or minimum value of a function.
3. Electrical engineering: Quadratic functions are used in electrical engineering to model the behavior of electrical circuits.

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

A: To determine the direction of the parabola's opening in vertex form, you can look at the sign of the coefficient $a$. If $a$ is positive, the parabola opens upward. If $a$ is negative, the parabola opens downward.

Q: Can I use quadratic functions to model real-world data?

A: Yes, quadratic functions can be used to model real-world data. However, it's essential to ensure that the data follows a quadratic pattern before using a quadratic function to model it.

Q: What are some common mistakes to avoid when working with quadratic functions?

A: Some common mistakes to avoid when working with quadratic functions include:

1. Failing to factor out the coefficient of $x^2$ when converting to vertex form.
2. Not adding and subtracting the square of half the coefficient of $x$ inside the parentheses.
3. Not simplifying the expression after completing the square.

Conclusion

In this article, we have addressed some of the most common questions related to quadratic functions, including their vertex form, graphing, and applications. By understanding these concepts, you can better work with quadratic functions and apply them to real-world problems.

Key Takeaways

• The vertex form of a quadratic function is $f(x) = a(x - h)^2 + k$.
• To convert a quadratic function to vertex form, you need to complete the square.
• The vertex of a quadratic function represents the maximum or minimum point of the parabola.
• Quadratic functions have numerous real-world applications, including projectile motion, optimization problems, and electrical engineering.

Practice Problems

1. Convert the quadratic function $f(x) = 2x^2 + 12x$ to vertex form.
2. Graph the quadratic function $f(x) = 3(x - 2)^2 + 1$.
3. Use a quadratic function to model the trajectory of a projectile under the influence of gravity.

Solutions

1. $f(x) = 2(x + 6)^2 - 36$
2. The graph of the quadratic function $f(x) = 3(x - 2)^2 + 1$ is a parabola that opens upward with a vertex at $(2, 1)$.
3. The quadratic function $f(x) = -16x^2 + 64x$ can be used to model the trajectory of a projectile under the influence of gravity.

Additional Resources

For more practice problems and solutions, check out the following resources:

• Khan Academy: Quadratic Functions
• Mathway: Quadratic Functions
• Purplemath: Quadratic Functions