AP Precalculus -1.1 Change in Tandem- FRQ Exam Style Questions - Effective Fall 2023
AP Precalculus -1.1 Change in Tandem- FRQ Exam Style Questions – Effective Fall 2023
AP Precalculus -1.1 Change in Tandem- FRQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.
▶️ Answer/Explanation
Detailed solution
Part 1: Transformation from \(y = f(2x + 1)\) to \(y = f(3x + 2)\)
To transform the graph, apply a horizontal translation left by \(\frac{1}{2}\) unit followed by a horizontal compression by a factor of \(\frac{2}{3}\).
Reasoning: Shifting left by \(\frac{1}{2}\) changes \(x \to x + \frac{1}{2}\), giving \(f(2(x+0.5)+1) = f(2x+2)\). Compressing by \(\frac{2}{3}\) changes \(x \to \frac{3}{2}x\), resulting in \(f(2(\frac{3}{2}x)+2) = f(3x+2)\).
To transform the graph, apply a horizontal translation left by \(\frac{1}{2}\) unit followed by a horizontal compression by a factor of \(\frac{2}{3}\).
Reasoning: Shifting left by \(\frac{1}{2}\) changes \(x \to x + \frac{1}{2}\), giving \(f(2(x+0.5)+1) = f(2x+2)\). Compressing by \(\frac{2}{3}\) changes \(x \to \frac{3}{2}x\), resulting in \(f(2(\frac{3}{2}x)+2) = f(3x+2)\).
Part 2: Two ways to obtain \(y = f(2x + 1)\) from \(y = f(x)\)
Way 1 (Shift then Scale):
First, apply a horizontal translation left by 1 unit to get \(y = f(x+1)\). Then, apply a horizontal compression by a factor of \(\frac{1}{2}\). This replaces \(x\) with \(2x\), resulting in \(y = f(2x+1)\).
First, apply a horizontal translation left by 1 unit to get \(y = f(x+1)\). Then, apply a horizontal compression by a factor of \(\frac{1}{2}\). This replaces \(x\) with \(2x\), resulting in \(y = f(2x+1)\).
Way 2 (Scale then Shift):
First, apply a horizontal compression by a factor of \(\frac{1}{2}\) to get \(y = f(2x)\). Then, apply a horizontal translation left by \(\frac{1}{2}\) unit. This replaces \(x\) with \(x+\frac{1}{2}\), resulting in \(y = f(2(x+0.5)) = f(2x+1)\).
First, apply a horizontal compression by a factor of \(\frac{1}{2}\) to get \(y = f(2x)\). Then, apply a horizontal translation left by \(\frac{1}{2}\) unit. This replaces \(x\) with \(x+\frac{1}{2}\), resulting in \(y = f(2(x+0.5)) = f(2x+1)\).