1. Graphing f(x) = ax^3
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Start by graphing y=ax^2 when a=1. Then, add graphs of the function considering different values of a. What is the effect of the parameter for each of the cases mentioned?
See solution.
We can first graph the function y=ax^2 for the simplest case — this is when a=1. Then, we can add graphs of the function considering other values of a. Graphing the functions together will allow us to see the effects of a. We will do this for the different cases mentioned in the exercise.
For this analysis we will study the effects of using a=0.25. To do this we can graph y_1=x^2 and y_2=0.25x^2.
As we can see, the factor a=0.25 reduces the value of the original function at every point, shrinking the graph vertically and taking it to be closer to the x-axis.
As we can see, the factor a=2 increases the value of the original function at every point, stretching the graph vertically and taking it away from the x-axis.
For this case, it will be useful to graph three functions together to visualize all of the transformations taking place. We can graph y_1=x^2, y_2=- x^2 and y_3=- 0.25x^2.
As we can see, having a negative a parameter causes a reflection in the x-axis. Furthermore, having a=- 0.25 decreases in absolute value the y-values of y_2=- x^2, shrinking the graph vertically and taking it closer to the x-axis.
Once more, it is useful to graph three functions together to visualize all of the transformations taking place. We will graph y_1=x^2, y_2=- x^2, and y_3=- 2 x^2 this time.
As we already mentioned above, having a negative a parameter causes a reflection in the x-axis. However, now that we are using a=- 2, the parameter increases in absolute value the y-values of the function y_2=- x^2, stretching the graph vertically and taking it away from the x-axis.
From the results we found above we can summarize the effects of the parameter a in the graph of y=ax^2