Vertex form is a mathematical function that can be used to represent a variety of different situations. For example, it can be useful for optimization problems and graphing quadratic functions. Therefore, knowing how to write a given equation in vertex form is an important part of understanding and using this form.
In this article, we’ll discuss the first step in writing f(x) = 6×2 + 5 – 42x in vertex form and provide an overview of other important topics related to vertex form.
Definition Of Vertex Form
The vertex form is a way to express a quadratic function of the form f(x) = a(x-h)^2 + k, where (h,k) represents the coordinates of the vertex of the parabola. This form offers insight into the vertex point and the direction of the parabola’s opening, making it a useful tool in analyzing quadratic functions.
The first step in writing f(x) = 6x^2 + 5 – 42x in vertex form is to complete the square by factoring the coefficient of x^2, which is 6. This can be achieved by dividing the equation by 6 to get f(x) = x^2 – 7x + 5/6.
Next, we need to add and subtract the square of half of the coefficient of x, which is (-7/2)^2 = 49/4. This gives us the equation f(x) = (x-7/2)^2 – 59/12 in vertex form, where the vertex point is (7/2, -59/12).
Understanding vertex form and how to convert standard form equations to vertex form is essential in solving quadratic problems and graphing parabolas.
Pro Tip: When converting a quadratic function from standard form to vertex form, always start by factoring the coefficient of x^2 and completing the square. This will make it easier to identify the vertex point and the direction of the parabola’s opening.
Advantages of Writing Equations in Vertex Form
The main advantage of writing equations in vertex form is that it allows us to easily identify the vertex, a key point on the graph that can provide important information about the function, such as its maximum or minimum value.
To write an equation in vertex form, the first step is to complete the square by adding and subtracting a constant term equal to half of the coefficient of the x-term squared. For example, in the equation f(x) = 6x^2 + 5 – 42x, the first step to write it in vertex form is to factor out the coefficient of the x^2 term, which is 6. This gives us f(x) = 6(x^2 – 7x/3) + 5. To complete the square, we need to add and subtract (7/3)^2 = 49/9, giving us f(x) = 6(x – 7/6)^2 – 97/3 in vertex form. This form reveals that the vertex is at (7/6, -97/3), and the function has a minimum value of -97/3.
Writing equations in vertex form can be a useful tool for solving problems and understanding the behavior of functions on the coordinate plane.
How to Convert f(x) = 6x^2 + 5 – 42x into Vertex Form
The first step in writing f(x) = 6x^2 + 5 – 42x in vertex form is to complete the square. We can use the following steps to convert the given equation into vertex form:
- Group the x^2 and x terms together: f(x) = 6x^2 – 42x + 5
- Factor out the coefficient of the x^2 term: f(x) = 6(x^2 – 7x) + 5
- To complete the square, take half of the coefficient of the x term (-7/2) and square it: (-7/2)^2 = 49/4
- Add and subtract the value obtained in step 3 inside the parenthesis: f(x) = 6(x^2 – 7x + 49/4 – 49/4) + 5
- Simplify and rearrange the terms: f(x) = 6[(x – 7/2)^2 – 49/4] + 5
- The vertex form of the equation is y = a(x – h)^2 + k, where (h, k) is the vertex. Comparing with the above equation, we get vertex form as f(x) = 6(x – 7/2)^2 – 167.
By completing the square, we can easily obtain the vertex form of the quadratic equation, which gives us important information about the vertex, axis of symmetry, and the maximum or minimum value of the quadratic function.
What Is The First Step In Writing f(x) = 6×2 + 5 – 42x in vertex form?
Writing an equation in vertex form is useful when graphing any equation. It allows you to quickly identify the graph’s vertex and axis of symmetry. So if you want to find out how to write f(x) = 6×2 + 5 – 42x in vertex form, the first step will be to complete the square. Let’s explore how to do that in more detail.
What is Completing the Square?
Completing the square is a method used in algebra to solve quadratic equations. For example, the first step in writing f(x) = 6x² + 5 – 42x in vertex form using the completing the square method is to isolate the x-terms and constant terms, and then divide the coefficient of the x-terms by 2 and square the result.
In this case, we can start as follows:
f(x) = 6x² – 42x + 5
= 6(x² – 7x) + 5
= 6(x² – 7x + 24.5 – 24.5) + 5
= 6((x-3.5)² – 24.5) + 5
Therefore, the vertex form of the equation is f(x) = 6(x-3.5)² – 139, and the vertex of the parabola is (3.5, -139).
Completing the square is a useful technique in solving quadratic equations and finding the vertex form of the equation.
How to Complete the Square for Quadratic Equations
The first step in completing the square for quadratic equations is to isolate the constant term and the coefficient of the x-term. For example, in the equation f(x) = 6x^2 + 5 – 42x, the constant term is 5 and the coefficient of the x-term is -42. To complete the square, we need to manipulate the equation so that it looks like f(x) = a(x – h)^2 + k, where (h, k) is the vertex of the parabola.
Here are the steps to follow:
Step – Equation
- f(x) = 6x^2 + 5 – 42x
- f(x) = x^2 – (7/3)x + 5/6
- f(x) + (49/36) = (x – 7/6)^2
- f(x) + (49/36) = x^2 – (7/3)x + (49/36) = (x – 7/6)^2
- f(x) = (x – 7/6)^2 – (49/36)
Therefore, the vertex form of the given equation is f(x) = (x – 7/6)^2 – (49/36).
Applying Completing the Square to Convert f(x) = 6x^2 + 5 – 42x to Vertex Form
The first step in converting f(x) = 6x^2 + 5 – 42x to vertex form using completing the square method is to factor out the coefficient of x^2, which is 6. The resulting expression will be f(x) = 6(x^2 – 7x) + 5.
The next step is to take half of the coefficient of x, which is -7, and square it. This will result in 24. Add and subtract 24 inside the parentheses to maintain the equality of the expression. f(x) = 6(x^2 – 7x + 24 – 24) + 5
Now, simplify the terms in the parentheses by factoring it as a perfect square trinomial. f(x) = 6[(x – 7/2)^2 – 1/4 x 24] + 5
Simplify the expression by multiplying 6 with -6 to get -36. Finally, write the expression in vertex form. f(x) = 6(x – 7/2)^2 – 219/4. The vertex is at (7/2, -219/4).
Completing the square method is a useful tool to convert quadratic equations to vertex form and find the vertex of the parabola. It is essential in various applications such as optimization and solving quadratic inequalities.
Finding The Vertex
Finding the vertex of a quadratic equation is a critical step in rewriting a quadratic equation in vertex form. In this article, we will discuss the first step for writing f(x) = 6×2 + 5 – 42x in vertex form. Then, we will discuss how to identify the vertex of a quadratic equation, the steps to take from there, and why it is important. Read on to learn more about finding the vertex of a parabola in an equation.
Definition of Vertex and its Significance
The vertex is the highest or lowest point on a parabolic curve, depending on the direction it opens. It is identified as (h, k), where h is the horizontal axis value of the vertex, and k is the vertical axis value of the vertex.
To find the vertex of a quadratic equation, such as f(x) = 6x² + 5 – 42x, the first step is to complete the square by factoring out the coefficient of x² and adding and subtracting (b/2a)². This will convert the equation to vertex form, y = a(x – h)² + k, where (h, k) is the vertex.
The vertex is significant because it provides valuable information about the parabolic function, including its highest or lowest point, the direction it opens, and the axis of symmetry. It also allows us to graph the function accurately and understand its behavior.
Pro tip: Understanding the concept of the vertex is crucial in solving and graphing quadratic equations. Practice using different equations to improve your understanding.
How to Find the Vertex of Quadratic Equations in Vertex Form
The vertex form of a quadratic equation is y = a(x-h)2 + k, where (h,k) represents the vertex of the parabola. To find the vertex of a quadratic equation in vertex form, follow these steps:
Identify the values of a, h, and k from the equation.
The vertex is located at the point (h,k).
Therefore, the first step in writing f(x) = 6×2 + 5 – 42x in vertex form would be to complete the square by factoring out the leading coefficient (6) from the first two terms, leaving a perfect square in parentheses. The equation would then become f(x) = 6(x2 – 7x + ) + 5. To complete the square, we need to add and subtract (7/2)2 = 49/4 inside the parentheses to get f(x) = 6(x – 7/2)2 – 127/2 in vertex form. Therefore, the vertex of this parabola is (7/2, -127/2).
Remember, to find the vertex of a quadratic equation in vertex form, it’s essential to identify the values of a, h, and k and use the formula (h,k) to locate the vertex of the parabola.
Finding the Vertex of f(x) = 6x^2 + 5 – 42x in Vertex Form
The first step in writing f(x) = 6x^2 + 5 – 42x in vertex form is to complete the square. The vertex form of a quadratic function is given by f(x) = a(x – h)^2 + k, where (h, k) is the vertex of the parabola. To obtain this form, we need to rewrite the given function by adding and subtracting a constant term that allows us to factor the squared term. Here are the steps to follow:
Steps
1. Group the x^2 and x terms: f(x) = 6x^2 – 42x + 5
2. Factor out the coefficient of the x^2 term: f(x) = 6(x^2 – 7x) + 5
3. Complete the square by adding and subtracting (7/2)^2 = 49/4: f(x) = 6(x^2 – 7x + 49/4 – 49/4) + 5
4. Factor the squared term and simplify: f(x) = 6(x – 7/2)^2 – 151/4
Therefore, the vertex form of f(x) = 6x^2 + 5 – 42x is f(x) = 6(x – 7/2)^2 – 151/4, and the vertex of the parabola is (7/2, -151/4).
Common Mistakes To Avoid
Writing a quadratic equation in vertex form can involve a few steps, but if done correctly you can easily find the vertex of the equation efficiently. Unfortunately, making mistakes while writing the equation in vertex form is often easy.
Let us look at some of the most common mistakes people make while writing an equation in vertex form and how to avoid them.
Common Errors in Completing the Square
The first step in completing the square for a quadratic equation is isolating the constant term on one side of the equation and the variable terms on the other. For example, to write f(x) = 6x² + 5 – 42x in vertex form, we need to follow the steps of completing the square:
Step 1: Group the variable terms together and factor out the common coefficient of x²:
f(x) = 6x² – 42x + 5
= 6(x² – 7x) + 5
Step 2: To complete the square, we need to add and subtract the square of half of the coefficient of x.
= 6(x² – 7x + (7/2)² – (7/2)²) + 5
= 6(x – 7/2)² – 79/2
Therefore, f(x) = 6(x – 7/2)² – 79/2 is the vertex form of the given quadratic equation.
Common errors to avoid when completing the square include:
- Forgetting to factor out the common coefficient of x².
- Making mistakes when adding and subtracting the square of half the coefficient of x.
- Failing to simplify the equation into the proper vertex form.
Pro Tip: Double-check your work at each step to avoid making careless mistakes.
Common Errors in Finding the Vertex
When finding the vertex of a quadratic equation, there are common errors that students should avoid to get the correct answer.
Some of these mistakes include:
- Forgetting to divide the coefficient of the x-term by 2 before squaring it to find the vertex’s x-coordinate.
- Misinterpreting the sign of the vertex’s x-coordinate, which is the opposite of the x-term’s coefficient.
- Incorrectly identifying the vertex’s coordinates by mistakenly taking the maximum or minimum y-value in the equation as the vertex.
To write f(x) = 6x^2 + 5 – 42x in vertex form, the first step is to complete the square by adding and subtracting a constant value. This will convert the equation into the form f(x) = a(x – h)^2 + k, where (h, k) is the vertex’s coordinates. Understanding the correct steps and avoiding the common errors will help students find the vertex accurately and efficiently.
Tips to Avoid Mistakes in Writing Quadratic Equations in Vertex Form
The first step in writing f(x) = 6×2 + 5 – 42x in vertex form is to complete the square. This involves rearranging the equation to isolate the x-terms, dividing them by the coefficient of the squared term (6 in this case), adding a constant term to both sides of the equation, and factoring the resulting perfect square trinomial.
Common mistakes to avoid when writing quadratic equations in vertex form include forgetting to divide the x-terms by the coefficient of the squared term, making errors in sign when adding the constant term, and not fully factoring the perfect square trinomial.
To avoid these mistakes, it’s important to go through each step carefully and double-check each calculation. It can also be helpful to practice with sample problems and seek feedback from a teacher or tutor. Pro tip: Remember that the vertex form of a quadratic equation is y = a(x – h)2 + k, where (h, k) is the coordinate of the vertex.
