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Parabolas

Q:  Consider the quadratic equation: y = 2x² -14x + 20:

(a) Put the equation in general form & find the vertex

(b) Put the equation in standard form & find the vertex

A:

Part (a):

Good news… General form: y = ax² + bx + c

Our equation is already in general form!  y = 2x² -14x + 20 is it.

Now, how do we find the vertex?  The x-coordinate of the vertex when in general form is -b/2a

So, x = -b/2a = -(-14)/(2*2) = 14/4 = 7/2

Now, to find the y-coordinate of the vertex, we plug in x = 7/2 to our equation and solve:

y = 2x² -14x + 20

y = 2(7/2)² -14(7/2) + 20

y = 2(49/4) – 98/2 + 20

y = 49/2 – 49 + 20 = -9/2

So, the vertex is (7/2, -9/2)

(b)  Something new now… Standard form of a quadratic is: y = a(x – h)² + k

To get to this form, we must complete the square:

We start with the equation:

y = 2x² -14x + 20

Factor a 2 out from the right like so:

y = 2(x² – 7x) + 20

Now, take the number that is multiplying the “x” and divide it by 2.  In our case, this would be -7.  So, we have -7/2.  Now, square this number: (-7/2)² = 49/4

We are going to add 49/4 to the right side, so we are also going to subtract it from the right side… However, it is a little more complicated than that.  I want to add “49/4” inside the parenthesis on the right side.  Notice that everything inside the parenthesis is being multiplied by 2??  So, if I add 49/4 inside the parenthesis, I have to take off “2 times 49/4” in order to balance it.  Does this make sense?  Watch:

y = 2(x² – 7x) + 20

y = 2(x² – 7x + 49/4) + 20 – 2*(49/4)

So I added something and subtracted the same thing… That is total legal – it is like adding nothing!  Remember, it is that “2” in front of the parenthesis that made this process a little more difficult, not all equations will have that problem.

OK, now simplify the back part:

y = 2(x² – 7x + 49/4) + 20 – 2*(49/4)

y = 2(x² – 7x + 49/4) + 20 – 49/2

y = 2(x² – 7x + 49/4) – 9/2

Now we can factor the junk in the parentheses… If done correctly, it will always factor into (x – half middle term)(x – half middle term)… Remember how we took half of the middle term earlier? Which turned out to be -7/2?  So, we can factor like so:

y = 2(x – 7/2)(x – 7/2) – 9/2

y = 2(x – 7/2)² – 9/2

There is our standard form!  y = a(x – h)² + k where the vertex is (h, k)

The vertex of our parabola is: (7/2, -9/2)

Good news since we got the same answer for part (a) and part (b).  We must have done it right!

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