# How can I graph a parabola?

Feb 7, 2015

Note: Since the topic is related to projectile motion, I will set an example that looks like an parabola; if you need help understanding how parabolas look like and their key components, you will probably need to reference algebra mathematics in order to understand trajectory.

In most trajectory problems, the best thing to do is first draw a simple picture. For example, let's start with a cannon shooting off a ball at some trajectory.

As the ball shoots off at a curvature, you will notice that it looks like an upside down parabola. First thing first, draw your coordinate system (+y going up and +x going right). The origin is the starting point of the cannon (assume no initial height of the cannon) going at some velocity diagonally, with an angle respect to a horizontal line.

If you use gravity as your acceleration in the y-direction, you can use the equations

(1) $\Delta y = - \frac{1}{2} \left(g\right) {\left(\Delta t\right)}^{2} + {v}_{\text{i,y}} \left(\Delta t\right)$ and
(2) ${v}_{\text{x}} = \frac{\Delta x}{\Delta t}$

to get certain aspects of time, displacement, and velocity. These forms are called the parameters of x and y, in which time is interchangeable between both directions (x and y). So, you can substitute the change in time ($\Delta t$) into another to solve for a specific problem (which will give you the Cartesian form of the path on picture). If the initial time is zero, then it would equal the final time.

Eq. 1 shows a similar parabola equation drawn upside down. Velocity in the y-direction is dependent on gravity, which is negative by direction not magnitude, as shown with the equation

(3) ${v}_{f , y} = - g \left(\Delta t\right) + {v}_{i , y}$ (linear form).

Unless there is acceleration in the x-direction, eq. 2 states that velocity is constant as the ball flies in the air, meaning that it is also linear.

The equations above are great for solving for the maximum displacement, such as the maximum height ($\Delta y$) of the ball and the distance traveled until the ball hits the ground ($\Delta x$), and much more.

Keep practicing and you will see a pattern as you do these problems.