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WHAT IS THE FIRST STEP IN WRITING F(X) = 6X2 + 5 – 42X IN VERTEX FORM? FACTOR 6 OUT OF EACH TERM. FACTOR 6 OUT OF THE FIRST TWO TERMS. WRITE THE FUNCTION IN STANDARD FORM. WRITE THE TRINOMIAL AS A BINOMIAL SQUARED.: Everything You Need to Know
What is the First Step in Writing f(x) = 6x2 + 5 – 42x in Vertex Form? Factor 6 Out of Each Term. Factor 6 Out of the First Two Terms. Write the Function in Standard Form. Write the Trinomial as a Binomial Squared?
Step 1: Review the Given Function
To begin, we need to examine the given function, which is f(x) = 6x2 + 5 – 42x. We are tasked with rewriting this function in vertex form, factoring 6 out of each term, factoring 6 out of the first two terms, writing the function in standard form, and expressing the trinomial as a binomial squared. To tackle this, we need to break down the steps involved in each process.Step 2: Factor 6 Out of Each Term
The first step is to factor 6 out of each term in the function f(x) = 6x2 + 5 – 42x. To do this, we can use the distributive property of multiplication to rewrite each term. We start by dividing each term by 6:- 6x2 ÷ 6 = x2
- 5 ÷ 6 = 5/6
- –42x ÷ 6 = –7x
By factoring 6 out of each term, the function becomes f(x) = 6(x2 – 7x) + 5/6.
Step 3: Factor 6 Out of the First Two Terms
Now that we have factored 6 out of each term, we can factor 6 out of the first two terms. We can do this by factoring the greatest common factor (GCF) of the first two terms, which is 6(x). We can rewrite the function as f(x) = 6(x(x – 7) + 5/6).Step 4: Write the Function in Standard Form
The standard form of a quadratic function is f(x) = ax2 + bx + c, where a, b, and c are constants. We can rewrite the function f(x) = 6(x(x – 7) + 5/6) in standard form by distributing the 6:| Term | Expression |
|---|---|
| First term | 6x(x – 7) |
| Second term | 6(5/6) |
Distributing the 6, we get f(x) = 6x2 – 42x + 5.
Step 5: Write the Trinomial as a Binomial Squared
The next step is to write the trinomial as a binomial squared. To do this, we need to identify a perfect square trinomial, which has the form (a – b)2 = a2 – 2ab + b2. We can rewrite the trinomial 6x2 – 42x + 5 as (3x – 7)2 by using the formula:- (3x – 7)2 = (3x)2 – 2(3x)(7) + (–7)2
- (3x – 7)2 = 9x2 – 42x + 49
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However, this is not equal to the original trinomial. We can try a different binomial squared, (3x – 4)2, but this also does not equal the original trinomial. Upon re-examining the trinomial, we can see that it is not a perfect square trinomial and cannot be written as a binomial squared.
What is the first step in writing f(x) = 6x2 + 5 – 42x in vertex form? Factor 6 out of each term. Factor 6 out of the first two terms. Write the function in standard form. Write the trinomial as a binomial squared.
serves as a crucial milestone in the journey of mathematical optimization. The vertex form of a quadratic function is a powerful tool for understanding the behavior of the function, and it is essential to master the techniques for converting a quadratic function from standard form to vertex form.
Understanding the Problem
When we are given a quadratic function in standard form, such as f(x) = 6x2 + 5 – 42x, our goal is to rewrite it in vertex form, which is f(x) = a(x-h)2 + k. To achieve this, we need to factor out the coefficient of the x2 term, which is 6, and then proceed with the necessary steps to write the function in vertex form.Factoring 6 out of Each Term
The first step in writing the function in vertex form is to factor 6 out of each term. This involves dividing each term by 6, which gives us f(x) = 6(x2) + (5/6) – (42/6)x. However, this is not the standard form that we need. To proceed, we need to factor 6 out of the first two terms, which will allow us to write the function in a more convenient form.Factoring 6 out of the First Two Terms
To factor 6 out of the first two terms, we can rewrite the function as f(x) = 6(x2 + (5/36) – (7/3)x). This allows us to see that the first two terms can be factored as a perfect square trinomial. By factoring 6 out of the first two terms, we have taken an important step towards rewriting the function in vertex form.Writing the Function in Standard Form
Now that we have factored 6 out of the first two terms, we can write the function in standard form. This involves expanding the perfect square trinomial and combining like terms. The resulting function is f(x) = 6(x - 7/6)2 + 25/36.Writing the Trinomial as a Binomial Squared
To write the trinomial as a binomial squared, we need to complete the square. This involves adding and subtracting a constant term to create a perfect square trinomial. The resulting function is f(x) = 6(x - 7/6)2 + 25/36, which can be rewritten as f(x) = 6(x - 7/6)2 + 25/36.Comparison of Methods
| Method | Steps | Complexity | | --- | --- | --- | | Factoring 6 out of each term | 1 step | Low | | Factoring 6 out of the first two terms | 1 step | Medium | | Writing the function in standard form | 2 steps | Medium | | Writing the trinomial as a binomial squared | 2 steps | High |Advantages and Disadvantages
| Method | Advantages | Disadvantages | | --- | --- | --- | | Factoring 6 out of each term | Easy to understand | Limited applicability | | Factoring 6 out of the first two terms | More applicable | Requires additional steps | | Writing the function in standard form | Convenient for calculations | May require additional steps | | Writing the trinomial as a binomial squared | Powerful tool for optimization | Requires advanced mathematical techniques |Conclusion
In conclusion, the first step in writing f(x) = 6x2 + 5 – 42x in vertex form is to factor 6 out of each term. However, this is not the only method, and we can also factor 6 out of the first two terms to achieve the same result. By understanding the different methods and their advantages and disadvantages, we can choose the most suitable approach for our specific problem. Ultimately, mastering the techniques for converting a quadratic function from standard form to vertex form requires practice and patience, but the rewards are well worth the effort.| Method | Steps | Complexity |
|---|---|---|
| Factoring 6 out of each term | 1 | Low |
| Factoring 6 out of the first two terms | 1 | Medium |
| Writing the function in standard form | 2 | Medium |
| Writing the trinomial as a binomial squared | 2 | High |
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