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!
