Solved Examples: Graphing Power Functions and Their Transformations

📚 Quick Study Guide

• 📈 Power Function Form: A power function is of the form $f(x) = ax^n$, where $a$ is a constant and $n$ is a real number.
• 🔍 Basic Graphs: Understand the basic shapes of $y = x^2$ (parabola), $y = x^3$ (cubic), $y = x^{-1} = \frac{1}{x}$ (hyperbola), and $y = x^{\frac{1}{2}} = \sqrt{x}$ (square root).
• ↔️ Horizontal Shifts: $f(x - h)$ shifts the graph $h$ units to the right if $h > 0$ and $|h|$ units to the left if $h < 0$.
• ↕️ Vertical Shifts: $f(x) + k$ shifts the graph $k$ units up if $k > 0$ and $|k|$ units down if $k < 0$.
• 💫 Vertical Stretch/Compression: $af(x)$ stretches the graph vertically by a factor of $|a|$ if $|a| > 1$ and compresses it if $0 < |a| < 1$. If $a < 0$, it also reflects the graph over the x-axis.
• 🔄 Horizontal Stretch/Compression: $f(bx)$ compresses the graph horizontally by a factor of $|b|$ if $|b| > 1$ and stretches it if $0 < |b| < 1$. If $b < 0$, it also reflects the graph over the y-axis.
• 💡 Reflection over x-axis: $-f(x)$ reflects the graph over the x-axis.
• 📝 Reflection over y-axis: $f(-x)$ reflects the graph over the y-axis.

Practice Quiz

1. Which of the following is a power function?
   1. A. $f(x) = 2^x$
   2. B. $f(x) = x^3$
   3. C. $f(x) = \sin(x)$
   4. D. $f(x) = \log(x)$
2. What transformation does $f(x) + 3$ represent on the graph of $f(x)$?
   1. A. Shift 3 units to the right
   2. B. Shift 3 units to the left
   3. C. Shift 3 units up
   4. D. Shift 3 units down
3. What transformation does $f(x - 2)$ represent on the graph of $f(x)$?
   1. A. Shift 2 units to the right
   2. B. Shift 2 units to the left
   3. C. Shift 2 units up
   4. D. Shift 2 units down
4. What transformation does $2f(x)$ represent on the graph of $f(x)$?
   1. A. Vertical compression by a factor of 2
   2. B. Vertical stretch by a factor of 2
   3. C. Horizontal compression by a factor of 2
   4. D. Horizontal stretch by a factor of 2
5. What transformation does $f(0.5x)$ represent on the graph of $f(x)$?
   1. A. Horizontal compression by a factor of 0.5
   2. B. Horizontal stretch by a factor of 0.5
   3. C. Horizontal stretch by a factor of 2
   4. D. Horizontal compression by a factor of 2
6. What transformation does $-f(x)$ represent on the graph of $f(x)$?
   1. A. Reflection over the y-axis
   2. B. Reflection over the x-axis
   3. C. Vertical shift
   4. D. Horizontal shift
7. Which transformation of $y = x^2$ would result in the graph $y = (x+1)^2 - 3$?
   1. A. Shift left 1 unit, shift up 3 units
   2. B. Shift right 1 unit, shift down 3 units
   3. C. Shift left 1 unit, shift down 3 units
   4. D. Shift right 1 unit, shift up 3 units