C L Oates
27 March 2011
OCCC Math 1513-017
Sketching Quadratic Functions
using a Formula from Analytical Geometry
In order to sketch a quadratic function, we need to transform a function like f(x) = 2x2 - 3x + 1 into the form a(x - h)2 + k. For the completing the square mechanism to work in this situation, we need to work on an expression of the form 1x2 + bx + c (notice that the coefficient of x2 has to be 1). In order to put a function like
f(x) = 2x2 - 3x + 1
into the appropriate form, we need to factor out 2 (the coefficient of x2) from each term of the function. I like to factor the 2 out of the constant term, as well as the other terms, so that everything works the way we did it in class.
f(x) = 2 (x2 - 3/2 x + 1/2)
Now if we operate inside the parentheses, everything works as it did in the in-class example. Let's see the "magic constant" becomes
-3/2 divided by 2, and the square that, so we get
(-3/4)2 = +9/16 for the "magic constant."
Now we can complete the square inside the parentheses like this:
f(x) = 2 (x2 - 3/2 x + 9/16 - 9/16 + 1/2) [EQUATION 1]
We haven't changed the function, because we have added 9/16 and then immediately subtracted 9/16, and
9/16 - 9/16 = 0.
Now, the completing the square method requires that we regroup inside the parentheses like this:
f(x) = 2 [ (x2 - 3/2 x + 9/16) - 9/16 + 1/2]. [EQUATION 2]
This regrouping is allowable, because we can perform additive operations in any order we like.
Now, if we did our "magic constant" calculation correctly, the terms inside the parentheses above (not the square brackets, the *parentheses*) should form the perfect square
(x - 3/4)2 .
Is that true? FOIL it out:
(x - 3/4)2 = (x - 3/4) (x - 3/4) = x2 - 3/4 x - 3/4 x + 9/16 and
= x2 - 6/4 x + 9/16
= x2 - 3/2 x + 9/16 . <---Yes, that's the expression in parenthesis in EQUATION 2.
That lets us rewrite EQUATION 2 like this:
f(x) = 2 [ (x2 - 3/2 x + 9/16) - 9/16 + 1/2]. [EQUATION 2--repeated]
f(x) = 2 [ (x - 3/4)2 - 9/16 + 1/2]. [EQUATION 3]
Now we're a lot closer to the desired expression of the form "a(x - h)2 + k." We'll have to do a little cleaning up. First, let's turn the 1/2 into 8/16 (by multiplying 1/2 * 8/8) to get a common denominator for the fractions. That gives us
f(x) = 2 [ (x - 3/4)2 - 9/16 + 8/16]. [EQUATION 4] and adding -9/16 to 8/16, we have
f(x) = 2 [ (x - 3/4)2 - 1/16 ]. [EQUATION 5]
Now we can multiply both the terms in square brackets by the 2 we factored out at first to get
f(x) = 2 (x - 3/4)2 - 2/16 . [EQUATION 6]
That's the desired form, with a = 2, h = +3/4 (remember the form is a(x 'minus' h)^2 + k), and k = -1/8 ( = reduced -2/16).
The vertex is then (3/4, -1/8).
The axis of symmetry is the vertical line, x = 3/4.
Since the coefficient of the leading term of 2 (x - 3/4)^2 - 1/8 is positive, the parabola will open upward, so we'll have a *minimum* value of -1/8 (the y-value of the vertex) at x = 3/4 (the x-value of the vertex).
Clearer? Clear as mud? Let me know.