"Why = a(x - h)² + k?" -- EVERYTHING YOU EVER WANTED TO KNOW ABOUT A QUADRATIC ...
The Quadratic As A Model For Projections
|Contents:||Here's Everything In One Picture||Stuff You Already Know|
|The Constant in the Equations||The Quadratic Coefficient in the Equations|
|Ex. 1 Ex. 2||Ex. 3 and Maximum Height|
|Ex. 4 and What The Equation Doesn't Say||Ex. 5: Review of What the Equation/Model Says|
|Here's Everything In One Picture
Don't worry. It's explained below.
The quadratic is a ideal model for projection -- shooting, propelling, dropping stuff at a certain speed, from a certain height.
The formula s(t) = (a/2)t2 + v0t + s0
|Stuff You Already Know
Don't worry. This reminds you.
On this page, we are not using x and y as the variables. We are using t and s.
The s stands for height or displacement or DISTANCE ABOVE (OR BELOW) GROUND LEVEL.
The t stands for TIME.
Restated, this equation is a model for how a thing moves relative to the ground over a period of time.
Restated, the horizonal scale measures the TIME and the vertical scale measures the DISTANCE ABOVE OR BELOW GROUND.Quick Examples
The equation "s(t) = 2" says "The height above ground is always 2."
The equation "s(t) = 2t" says "The height above ground is twice the time." At starting time, time 0, the height is 0 -- the object is on the ground. After 1 second, the height is 2. After 2 seconds, the height is 4. After 10 seconds, the height is 20.
The equation "s(t) = s0 + v0t + (a/2)t2" says "The height above ground is the sum of the following 3 terms (features): the starting height, the starting velocity times the time, and half the acceleration times the square of the time.
|The Constant in the Equations
Look at the constant terms.
|The Linear Coefficient in the Equations
|The Quadratic Coefficient in the Equations
Look at the coefficient of the quadratic term.
The power on the variable causes the trajectory, path, of the thing to be a parabola, a quadratic function. It makes the path a U-shaped curve.
In every equation, the coefficient of the quadratic term is negative. Gravity is bringing the object to the ground even if it is shot upward. The U-shaped curve opens down. If there existed a case in which the more massive object were up, instead of down as on earth, the gravity would be positive and things would not fall but rise and the coefficient of the quadratic term would be positive.
|Ex. 3 and Maximum Height
|Ex. 4 and What The Equation Doesn't Say
|Ex. 5: Review of What the Equation/Model Says
& A Typical Application Question