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Leno4ka [110]
3 years ago
9

Write the function in vertex form. y = 2x² + x

Mathematics
2 answers:
Sliva [168]3 years ago
6 0
<h2>Steps:</h2>
  • Vertex Form (aka general form): y=a(x-h)^2+k , with (h,k) as the vertex.

So for this, we are going to be completing the square. Firstly, we have to factor out the 2 out of the right side of the equation to make the x² coefficient 1:

y=2(x^2+\frac{1}{2}x)

Next, we want to make the quantity inside of the parentheses a perfect square. To find the constant of this soon-to-be perfect square, divide the x coefficient by 2 and then square the quotient. In this case:

\frac{1}{2}\div \frac{2}{1}\\\\\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}\\\\\\(\frac{1}{4})^2=\frac{1^2}{4^2}=\frac{1}{16}

Now, we are going to be adding 1/16 inside of the parentheses. Now we want to cancel this quantity out (note: addition property of equality). To do this, we want to add the product of 1/16 and 2 (since 2 is multiplying with 1/16 on the right side) to the left side:

\frac{1}{16}\times\frac{2}{1}=\frac{2}{16}=\frac{1}{8}\\\\y+\frac{1}{8}=2(x^2+\frac{1}{2}x+\frac{1}{16})

Now that the quantity inside the parentheses is a perfect square, factor:

  • Tip: (x+y)^2=x^2+2xy+y^2

x^2+\frac{1}{2}x+\frac{1}{16}=(x+\frac{1}{4})^2\\\\y+\frac{1}{8}=2(x+\frac{1}{4})^2

Lastly, subtract both sides by 1/8:

y=2(x+\frac{1}{4})^2-\frac{1}{8}

<h2>Answer:</h2>

<u>In short, your vertex form is: y=2(x+\frac{1}{4})^2-\frac{1}{8}</u>

N76 [4]3 years ago
4 0

To start to solve this problem, we need to know what vertex form is. The vertex form of a parabola is. The vertex form of a parabola is a(x-h) + k, where k is the vertical shift, h is the horizontal shift, and a is the value that tells the stretch.


To start to solve this equation, we want to start to create a difference of two squares.

y = 2(x²+\frac{1}{2}x) We do this step to make the x² have a coefficient of 1

Now, we want to complete the square. To complete the square, we take 1/2 of the coefficient of x, and then square that.

1/2 * 1/2 = 1/4, and 1/4²=1/16

That means that we need to add 1/16 inside and outside the parenthesis.

We get:

y = 2(x²+1/2x + 1/16) - 1/16*2

We do -1/16*2 on the outside because since we added it inside the parenthesis, we need to take it away somewhere else (if that makes sense). The two is there because there is a two in front of the parenthesis.

We get:

y = 2(x+1/4)² - 1/8, by completing the square and simplifying, and this is the final answer.



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        a_1=4\\\\a_2=0\\\\a_3=3

Explanation:

The equation is:

      a_n=3a_{n-1}-5a_{n-2}-4a_{n-3}

and

      a_4=-7, a_5=-36, a_6=-85

<u>1. Subsittute n = 6 into the equation:</u>

      a_6=3a_{5}-5a_{4}-4a_{3}

      Now subsititute the known values:

       -85=3(-36)-5(-7)-4a_{3}

      You can solve for a_3

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<u />

<u>2. Substitute n = 5 into the equation:</u>

    a_5=3a_{4}-5a_{3}-4a_{2}

     Substitute the known values and solve for a_2

        -36=3(-7)-5(3)-4a_{2}\\\\4a_2=-21-15+36=0\\\\a_2=0

<u>3. Substitute n = 4 into the equation, subsitute the known values and solve for </u>a_1<u />

       a_4=3a_{3}-5a_{2}-4a_{1}

        -7=3(3)-5(0)-4a_{1}\\\\4a_1=9+7=16\\\\a_1=4

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