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# Week 9 in PreCalc 11

Something I’ve learned this week is about modeling. No – not the modeling one where you pose for a picture – what I’m talking about is creating a quadratic equation based on a word problem, like regarding revenue, money, finding numbers, projectile motions, etc.

There’s actually no need for me to explain what it is as it’s pretty straightforward. All you have to do is analyze the word problem carefully and try and make an equation (of course, with a graph!) based on the problem.

Now, let’s take a look at this problem.

Find two integers that has a sum of 16, and the greatest possible product.

two numbers that has a sum of 16. In other words, it can be written like this algebraically,

x + y = 16

which then, we can arrange into…

y = 16 – x

Now, let’s say that

$x = 1^{st} number$

and…

\$latex 16 – x = 2^{nd} number\$

Now, if we add those two, we should get a sum of 16.

Why? Well, since our first number is x, and we arranged the equation to become y=16-x, which then means that y has the same value as 16-x, then if you add

x and 16-x, you should get 16 as in the first equation you’re adding x and y.

Anyways, since we also need to find the greatest possible product, we’re gonna multiply them instead of adding, which we’ll get the equation…

$x (16 - x) = y$

Right now, it’s in the factored form, so it means we already have our x-intercepts or roots, which are:

x = 0 ; and x = 16

Now what? Well, we need to find the value of the vertex, because the x-value is the value of each two integers that we need to find and the y-value is the product that we also need to find.

we can find the x-value of the vertex in two ways: graphing and calculating.

For calculating, we just need to find the average, which is pretty easy.

0 + 16 = 16

16 / 2 = 8

And x = 0 is your vertex’ x-value.

Now, for the graphing, we just need to sketch it. but since we already have the x-value let’s do a sketch…

Now, how do we figure out the product? Well, just plug the x-value of vertex in your formula.

f(8) = 8 (16-8)

f(8) = 8 (8)

f(8) = 64

Well, there you have it!

Two integers that has a sum of 16 is 8, and 8 with the greatest possible product of 64!