Smile

Katia S.
My parabola is a smile. I found the image off of the internet. It is 5 centimeters across from one corner to the other and it is 1 3/4 centimeters tall (picture shown is the same image but a smaller size). The Guinness World Record for the longest continuous smile is currently held by Pastor Wally Holt from Illinois who smiled for 423 days straight.

To get my equation of the parabola in standard form, I put in the three points I knew (the vertex and each corner of the mouth) into Microsoft Excel and made a graph out of it.



My standard equation is y=0.28x². The 2E-16x+9E-16 shown on the graph equals about 0 so it can be ignored from the equation. If I work backwards with my equation, I can figure out the equation in factored form which is y= 0.28(x)(x). Because I know that the vertex is (0,0), I can discover the vertex form of the equation by filling in (h,k) in y-k=a(x-h)² with my vertex. So the vertex form is y-0=(x-0)².

The vertex of my graph is (0,0) and it is the minimum of the graph. It represents the bottom of the smile. The y-intercept is 0 and it is also the x-intercept and the vertex. To find the discriminant, I will fill in the equation b²-4ac. The discriminant is 0 so I only have 1 x-intercept. My only x-intercept is the same as the y-intercept and vertex. It is (0,0).

A person eating soup gets it on their mouth. It is 2 centimeters from the middle of the mouth. If the person is smiling, what is the height of the soup on the mouth?

.28x² .28(2)² .28(4) y=1.12 centimeters