Question

1. reinforce write your own description of a series of transformations and the function rule that matches that description.

Explanation:

Let's consider the parent function \(y = x^{2}\). First, we can perform a vertical stretch by a factor of \(2\). This means that for each \(y\)-value of the original function, we multiply it by \(2\). The rule for a vertical stretch of \(y = f(x)\) by a factor \(a\) is \(y=a\cdot f(x)\). So, after the vertical stretch, the function becomes \(y = 2x^{2}\).

Next, we can perform a horizontal shift to the right by \(3\) units. The rule for a horizontal shift of \(y = f(x)\) to the right by \(h\) units is \(y=f(x - h)\). So, substituting \(h = 3\) into our function \(y = 2x^{2}\), we get \(y=2(x - 3)^{2}\).

Finally, we can perform a vertical shift up by \(4\) units. The rule for a vertical shift of \(y = f(x)\) up by \(k\) units is \(y=f(x)+k\). Substituting \(k = 4\) into \(y=2(x - 3)^{2}\), we get the final function \(y=2(x - 3)^{2}+4\).

Answer:

The series of transformations: vertical stretch by a factor of \(2\), horizontal shift to the right by \(3\) units, and vertical shift up by \(4\) units. The function rule: \(y = 2(x - 3)^{2}+4\)