Recall the Vertex Form of a Quadratic Function

Oct 13, 2025 by windailys.com 114 views

Understanding the Problem

The given quadratic function is in the form of $y = ax^2 + bx + c$ , where $a = -2$ , $b = -16$ , and $c = -31$ . Our goal is to rewrite this equation in vertex form, which is given by $y = a(x - h)^2 + k$ , where $(h, k)$ represents the coordinates of the vertex of the parabola.

Recall the Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by:

y = a(x - h)^2 + k

where $(h, k)$ is the vertex of the parabola. To convert the given quadratic 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 the $x^2$ term, which is $-2$ . This gives us:

y = -2(x^2 + 8x) - 31

Next, we add and subtract the square of half the coefficient of the $x$ term, which is $8$ . Half of $8$ is $4$ , and the square of $4$ is $16$ . So, we add and subtract $16$ inside the parentheses:

y = -2(x^2 + 8x + 16 - 16) - 31

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

y = -2((x + 4)^2 - 16) - 31

Next, we distribute the $-2$ to the terms inside the parentheses:

y = -2(x + 4)^2 + 32 - 31

Simplifying the expression, we get:

y = -2(x + 4)^2 + 1

Identifying the Vertex

Comparing the final expression with the vertex form of a quadratic function, we can see that the vertex is given by $(h, k) = (-4, 1)$ .

Conclusion

In this article, we have learned how to convert a quadratic function from standard form to vertex form by completing the square. We have also identified the vertex of the parabola, which is given by $(h, k) = (-4, 1)$ . This process is useful in various applications, such as graphing quadratic functions and solving optimization problems.

Example

Suppose we want to graph the quadratic function $y = -2x^2 - 16x - 31$ . To do this, we can use the vertex form of the function, which is $y = -2(x + 4)^2 + 1$ . The vertex of the parabola is given by $(h, k) = (-4, 1)$ , which means that the parabola opens downward and has a minimum value of $1$ at the point $(-4, 1)$ .

Tips and Tricks

When completing the square, make sure to add and subtract the same value inside the parentheses.
When distributing the coefficient to the terms inside the parentheses, make sure to multiply each term by the coefficient.
When simplifying the expression, make sure to combine like terms.

Common Mistakes

Failing to add and subtract the same value inside the parentheses when completing the square.
Failing to distribute the coefficient to the terms inside the parentheses.
Failing to simplify the expression by combining like terms.

Real-World Applications

Graphing quadratic functions: By converting a quadratic function to vertex form, we can easily graph the function by plotting the vertex and using the symmetry of the parabola.
Solving optimization problems: By converting a quadratic function to vertex form, we can easily find the maximum or minimum value of the function by identifying the vertex.

Conclusion

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

A: The vertex form of a quadratic function is given by $y = a(x - h)^2 + k$ , where $(h, k)$ represents the coordinates of 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 the $x^2$ term, adding and subtracting the square of half the coefficient of the $x$ term, and then simplifying the expression.

Q: What is completing the square?

A: Completing the square is a process of rewriting a quadratic expression in the form of a perfect square trinomial. This involves adding and subtracting the same value inside the parentheses to create a perfect square.

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

A: To identify the vertex of a parabola, you need to look at the vertex form of the quadratic function. The vertex is given by $(h, k)$ , where $h$ is the value of $x$ and $k$ is the value of $y$ .

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

A: The vertex of a parabola represents the maximum or minimum value of the function. If the parabola opens upward, the vertex represents the minimum value. If the parabola opens downward, the vertex represents the maximum value.

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. By plotting the vertex and using the symmetry of the parabola, you can easily graph the function.

Q: How do I use the vertex form of a quadratic function to solve optimization problems?

A: To use the vertex form of a quadratic function to solve optimization problems, you need to identify the vertex of the parabola. The vertex represents the maximum or minimum value of the function, which is the solution to the optimization problem.

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

A: Some common mistakes to avoid when converting a quadratic function to vertex form include failing to add and subtract the same value inside the parentheses, failing to distribute the coefficient to the terms inside the parentheses, and failing to simplify the expression by combining like terms.

Q: Can I use technology to convert a quadratic function to vertex form?

A: Yes, you can use technology to convert a quadratic function to vertex form. Many graphing calculators and computer algebra systems can convert a quadratic function to vertex form.

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

A: To check your work when converting a quadratic function to vertex form, you need to plug the vertex form of the function back into the original function and simplify the expression. If the expression simplifies to the original function, then your work is correct.

Q: What are some real-world applications of converting quadratic functions to vertex form?

A: Some real-world applications of converting quadratic functions to vertex form include graphing quadratic functions, solving optimization problems, and modeling real-world phenomena.

Conclusion

In conclusion, converting a quadratic function to vertex form is a useful process that can be applied in various applications, such as graphing quadratic functions and solving optimization problems. By completing the square, we can rewrite the quadratic function in vertex form and identify the vertex of the parabola. This process requires attention to detail and careful simplification of the expression.