In the following activity, you will see [math]f\left(x\right)=a\cdot sin\left(b\cdot\left(x-d\right)\right)+c[/math][br]a, b, c and d are coefficients that play a role in altering the graph of the function. In this exploration, we will examine the impact that these coefficients have on the graph, and try to form a conjecture that is in alignment with our observations. In the activity, you will see [math]y=sin\left(x\right)[/math] against which the various transformations happen. The default values are a=1, b=1, c=0, d=0. Move the sliders to explore the transformation it effects on the graph, and write down your observation in space provided below.
Write down your observations about the impact [math]a[/math] has on the graph. You may also relate the value of the coefficient with the impact. List down as many impacts as you observe.
Write down your observations about the impact [math]b[/math] has on the graph. You may also relate the value of the coefficient with the impact. List down as many impacts as you observe.
Write down your observations about the impact [math]c[/math] has on the graph. You may also relate the value of the coefficient with the impact. List down as many impacts as you observe.
Write down your observations about the impact [math]d[/math] has on the graph. You may also relate the value of the coefficient with the impact. List down as many impacts as you observe.
Now replace [math]g(x)=sin(x)[/math] with [math]g\left(x\right)=x^2[/math] and observe the impact of the coefficients on this graph. State the impact of all the coefficients below.
Create your own function for g(x), predict the impact and observe if your prediction is accurate. State your prediction and the level of accuracy below.
Based on this exploration, form a conjecture that lists the impact that a, b, c and d have on the function's graph. The conjecture must ensure precision, listing the impact as well as the scope of the impact. Also, list down any constraints or assumptions you may have. After listing down the conjecture, select a mathematical approach to ascertian the truth of your conjecture.