**Write-Up #2 (Quadratics)**

*y = ax ^{2}+bx+c*

By Jaepil Han

Problem 2. Fix the values foraandb, varyc. Make at least 5 graphs on the same axes as you varyc.

y = ax^{2}+ bx + c

We want to focus on the patterns when the values of parameters a, b, and c vary. Here's the standard form of quadratic functions.

y = ax^{2}+ bx + cThe pattern of the quadratic equation when the values for a and b are fixed and the value for c varies from 3 to -3.

Here's the graphs using the Graphing Calculator.

When the parameter c varies from 3 to -3, the graph of the equation is vertically translated. Also, as you may noticed, the y-intercept of the equation is the same as the value of c. For example, the y-intercept of

y = xis 3 and that of^{2}+ x + 3y = xis -1.^{2}+ x - 1Also, here's the animation of the graphs when the value of c is varied from +5 to -5.

At this point, it seems that the pattern of the quadratic function when only the value for c varies is vertical translations of one of the quadratic functions. Or, we can express all those graphs are the translations of the quadratic equation y = x

^{2}+x+0.We might prove this is a translation by mapping. Since our students are not familiar with mapping stuff, we simply introduce it as a dilation of the quadratic equation y = x

^{2}+x+0.