Question

Write the function in vertex form.

y = 2x² + x

Answer 1

Steps:

• 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+(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:

`(1)/(2)/ (2)/(1)\\\\(1)/(2)*(1)/(2)=(1)/(4)\\\\\\((1)/(4))^2=(1^2)/(4^2)=(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:

`(1)/(16)*(2)/(1)=(2)/(16)=(1)/(8)\\\\y+(1)/(8)=2(x^2+(1)/(2)x+(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+(1)/(2)x+(1)/(16)=(x+(1)/(4))^2\\\\y+(1)/(8)=2(x+(1)/(4))^2`

Lastly, subtract both sides by 1/8:

`y=2(x+(1)/(4))^2-(1)/(8)`

Answer:

In short, your vertex form is:
`y=2(x+(1)/(4))^2-(1)/(8)`

Answer 2

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+
`(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.