Graphing Polynomials with Transformations: A Khan Academy Precalculus Lesson

Understanding the Parent Function

• 💡 The video focuses on graphing the function g(x) = 1/2 * (x - 3)^3 + 5.
• 🧠 This function is a transformed version of the parent function f(x) = x^3.
• 📈 Key points for f(x) = x^3 include (0,0), (1,1), (-1,-1), (2,8), and (-2,-8).

Applying Transformations

• 🔬 The transformation involves scaling, horizontal shifting, and vertical shifting.
• 🎯 Scaling by 1/2 transforms f(x) = x^3 into h(x) = 1/2 * x^3, which grows and shrinks slower.
• ➡️ Shifting right by 3 units is achieved by replacing x with (x - 3).
• ⬆️ Shifting up by 5 units is achieved by adding 5 to the function.

Graphing the Transformed Function

• 🧩 The point (0,0) on the parent function moves to (3,5) after the transformations.
• 📊 Points like (1, 1/2) and (-1, -1/2) are plotted relative to the new origin (3,5).
• 📉 Similarly, points like (2,4) and (-2,-4) are plotted relative to (3,5).
• ✨ The final graph of g(x) visually represents these combined transformations of the cubic parent function.