Rewrite the expression in vertex form

Oct 13, 2025 by magicalceylontours.com 59 views

Understanding the Basics of Vertex Form

In mathematics, the vertex form of a quadratic function is a way to express the function in a form that highlights its vertex, or turning point. This form is particularly useful for graphing and analyzing quadratic functions. The vertex form of a quadratic function is given by the equation:

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

where $(h, k)$ is the vertex of the parabola, and $a$ is a constant that determines the direction and width of the parabola.

Nick's Work: Converting a Quadratic Function to Vertex Form

Nick was given the quadratic function $p(x) = 17 + 42x - 7x^2$ and asked to convert it to vertex form. His work is shown below:

Step 1: Factor out the coefficient of the $x^2$ term

p(x) = -7x^2 + 42x + 17

Step 2: Factor out the coefficient of the $x$ term

p(x) = -7(x^2 - 6x) + 17

Step 3: Complete the square

To complete the square, Nick needed to find the value that would make the expression inside the parentheses a perfect square trinomial. He started by taking half of the coefficient of the $x$ term, which is $-6$ , and squaring it:

\left(\frac{-6}{2}\right)^2 = 9

Step 4: Rewrite the expression in vertex form

Using the value found in Step 3, Nick rewrote the expression in vertex form:

p(x) = -7(x^2 - 6x + 9) + 17 - 63

Simplifying the expression, Nick got:

p(x) = -7(x - 3)^2 + (-7)(9) + 17

p(x) = -7(x - 3)^2 - 49 + 17

p(x) = -7(x - 3)^2 - 32

Discussion

Nick's work shows that converting a quadratic function to vertex form involves several steps, including factoring out the coefficient of the $x^2$ term, factoring out the coefficient of the $x$ term, completing the square, and rewriting the expression in vertex form. By following these steps, Nick was able to convert the given quadratic function to vertex form.

Why is Vertex Form Important?

Vertex form is an important way to express quadratic functions because it highlights the vertex of the parabola. The vertex form is particularly useful for graphing and analyzing quadratic functions, as it allows us to easily identify the vertex and the direction and width of the parabola.

Real-World Applications of Vertex Form

Vertex form has many real-world applications, including:

Physics: Vertex form is used to model the motion of objects under the influence of gravity.
Engineering: Vertex form is used to design and analyze the shape of bridges and other structures.
Computer Science: Vertex form is used in computer graphics to create 3D models and animations.

Conclusion

In conclusion, converting a quadratic function to vertex form involves several steps, including factoring out the coefficient of the $x^2$ term, factoring out the coefficient of the $x$ term, completing the square, and rewriting the expression in vertex form. By following these steps, we can convert any quadratic function to vertex form and gain a deeper understanding of its properties and behavior.

Additional Resources

For more information on vertex form and quadratic functions, check out the following resources:

Mathway: A online math problem solver that can help you convert quadratic functions to vertex form.
Khan Academy: A free online resource that provides video lessons and practice exercises on quadratic functions and vertex form.
Wolfram Alpha: A powerful online calculator that can help you convert quadratic functions to vertex form and analyze their properties.

Practice Problems

Try converting the following quadratic functions to vertex form:

$p(x) = 2x^2 + 12x + 10$
$p(x) = -3x^2 - 6x + 2$
$p(x) = x^2 + 4x - 3$

Q&A: Vertex Form and Quadratic Functions

Q: What is vertex form?

A: Vertex form is a way to express a quadratic function in the form $p(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola, and $a$ is a constant that determines the direction and width of the parabola.

Q: Why is vertex form important?

A: Vertex form is important because it highlights the vertex of the parabola, which is the turning point of the function. This makes it easier to graph and analyze quadratic functions.

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

A: To convert a quadratic function to vertex form, follow these steps:

Factor out the coefficient of the $x^2$ term: Factor out the coefficient of the $x^2$ term from the quadratic function.
Factor out the coefficient of the $x$ term: Factor out the coefficient of the $x$ term from the quadratic function.
Complete the square: Complete the square by finding the value that would make the expression inside the parentheses a perfect square trinomial.
Rewrite the expression in vertex form: Rewrite the expression in vertex form using the value found in Step 3.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the turning point of the function, which is the point where the parabola changes direction.

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

A: To find the vertex of a parabola, use the formula $(h, k) = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$ , where $a$ , $b$ , and $c$ are the coefficients of the quadratic function.

Q: What is the significance of the vertex form in physics?

A: In physics, the vertex form is used to model the motion of objects under the influence of gravity. The vertex form is particularly useful for analyzing the trajectory of projectiles and the motion of objects under the influence of gravity.

Q: Can I use vertex form to solve quadratic equations?

A: Yes, you can use vertex form to solve quadratic equations. By setting the expression in vertex form equal to zero, you can solve for the roots of the equation.

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

A: To graph a quadratic function in vertex form, follow these steps:

Identify the vertex: Identify the vertex of the parabola, which is the point $(h, k)$ .
Determine the direction: Determine the direction of the parabola, which is determined by the value of $a$ .
Plot the vertex: Plot the vertex on the graph.
Plot the parabola: Plot the parabola using the vertex and the direction.

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

A: Yes, you can use vertex form to analyze the behavior of a quadratic function. By examining the vertex and the direction of the parabola, you can determine the behavior of the function.

Conclusion

In conclusion, vertex form is a powerful tool for analyzing and graphing quadratic functions. By understanding the properties of vertex form and how to convert a quadratic function to vertex form, you can gain a deeper understanding of the behavior of quadratic functions and their applications in physics and other fields.

Additional Resources

For more information on vertex form and quadratic functions, check out the following resources:

Mathway: A online math problem solver that can help you convert quadratic functions to vertex form.
Khan Academy: A free online resource that provides video lessons and practice exercises on quadratic functions and vertex form.
Wolfram Alpha: A powerful online calculator that can help you convert quadratic functions to vertex form and analyze their properties.

Practice Problems

Try converting the following quadratic functions to vertex form and analyzing their behavior:

$p(x) = 2x^2 + 12x + 10$
$p(x) = -3x^2 - 6x + 2$
$p(x) = x^2 + 4x - 3$

Use the steps outlined above to complete the square and rewrite the expression in vertex form.