Active Prelude to Calculus

Sketch an accurate graph of the transformation \(y = p(x) = -\frac{1}{2}f(x-1)+2\text{.}\) Write at least one sentence to explain how you developed the graph of \(p\text{,}\) and identify the point on \(p\) that corresponds to the original point \((-2,2)\) on the graph of \(f\text{.}\)

Sketch an accurate graph of the transformation \(y = q(x) = 2g(x+0.5)-0.75\text{.}\) Write at least one sentence to explain how you developed the graph of \(q\text{,}\) and identify the point on \(q\) that corresponds to the original point \((1.5,1.5)\) on the graph of \(g\text{.}\)

Is the function \(y = r(x) = \frac{1}{2}(-f(x-1) - 4)\) the same function as \(p\) in part (a) or different? Why? Explain in two different ways: discuss the algebraic similarities and differences between \(p\) and \(r\text{,}\) and also discuss how each is a transformation of \(f\text{.}\)

Find a formula for a function \(y = s(x)\) (in terms of \(g\)) that represents this transformation of \(g\text{:}\) a horizontal shift of \(1.25\) units left, followed by a reflection across the \(x\)-axis and a vertical stretch by a factor of \(2.5\) units, followed by a vertical shift of \(1.75\) units. Sketch an accurate, labeled graph of \(s\) on the following axes along with the given parent function \(g\text{.}\)