Rational function modeling

Hey kiddo, have you ever seen a fraction? Like 1/2, where there's a number on top and a number on the bottom, and they're separated by a line? That's called a rational expression or a fraction!

A rational function is a special type of fraction that has variables (like x or y) in the numerator or denominator, or both. For example, (3x + 2) / (4x - 1) is a rational function. The variables represent numbers that can change, so it's like having an infinite number of fractions all at once!

Now, let's say we want to use a rational function to model something in real life, like the height of a bouncing ball. We can use the formula h(t) = -16t^2 + vt + h0, where t is time in seconds, h(t) is the height of the ball at time t, v is the initial velocity of the ball, and h0 is the initial height of the ball.

But what if we want to know how long it takes for the ball to reach a certain height, like 10 feet? We can use a rational function to solve for t! We can set h(t) equal to 10 and solve for t, like this:

10 = -16t^2 + vt + h0
16t^2 - vt - h0 + 10 = 0

This is a quadratic equation, which means it has a squared variable. But we can simplify it by using the quadratic formula, which gives us:

t = (v +/- sqrt(v^2 + 64(h0 - 10)) / 32

Now, we have a rational function that tells us the time it takes for the ball to reach a specific height. We can input different values for v and h0 to see how they affect the time it takes for the ball to bounce!

So you see, rational function modeling is just using fractions with variables to represent real-life situations and solve problems. It's like having a magic formula that can tell us how things will behave! Cool, huh?