1.3 Transformations of functions

Function transformations are like giving your graph a makeover. You can shift, stretch, or flip it to create new functions. These changes let you model real-world scenarios more accurately, from population growth to sound waves.

Understanding transformations helps you see how changing parts of a function affects its graph. This skill is crucial for analyzing and predicting trends in various fields, from physics to economics. It's all about tweaking functions to fit real-life situations.

Function Transformations

Vertical and horizontal graph shifts

• Vertical shifts move the graph up or down without changing its shape
  • Add a positive constant $k$ to $f(x)$ to shift the graph up by $k$ units ($f(x) + k$)
  • Subtract a positive constant $k$ from $f(x)$ to shift the graph down by $k$ units ($f(x) - k$)
  • Example: $f(x) = x^2$ shifted up by 3 units becomes $g(x) = x^2 + 3$
• Horizontal shifts move the graph left or right without changing its shape
  • Replace $x$ with $(x - h)$ to shift the graph right by $h$ units ($f(x - h)$)
  • Replace $x$ with $(x + h)$ to shift the graph left by $h$ units ($f(x + h)$)
  • Example: $f(x) = \sin(x)$ shifted left by $\pi/2$ becomes $g(x) = \sin(x + \pi/2)$

Reflections and stretches of graphs

• Reflections flip the graph across the x-axis or y-axis
  • Multiply $f(x)$ by -1 to reflect the graph across the x-axis ($-f(x)$)
  • Replace $x$ with $-x$ to reflect the graph across the y-axis ($f(-x)$)
  • Example: $f(x) = x^3$ reflected across the x-axis becomes $g(x) = -x^3$
• Vertical stretches and compressions change the graph's height without affecting its width
  • Multiply $f(x)$ by a constant $a > 1$ to vertically stretch the graph by a factor of $a$ ($af(x)$)
  • Multiply $f(x)$ by a constant $0 < a < 1$ to vertically compress the graph by a factor of $a$ ($af(x)$)
  • Example: $f(x) = \sqrt{x}$ vertically stretched by a factor of 2 becomes $g(x) = 2\sqrt{x}$
• Horizontal stretches and compressions change the graph's width without affecting its height
  • Replace $x$ with $x/b$, where $b > 1$, to horizontally stretch the graph by a factor of $b$ ($f(x/b)$)
  • Replace $x$ with $bx$, where $0 < b < 1$, to horizontally compress the graph by a factor of $1/b$ ($f(bx)$)
  • Example: $f(x) = e^x$ horizontally compressed by a factor of 2 becomes $g(x) = e^{2x}$

Combining multiple graph transformations

• When applying multiple transformations, follow this order: reflections, stretches/compressions, horizontal shifts, vertical shifts
  • Perform reflections first to maintain the correct orientation of the graph
  • Apply stretches and compressions next to ensure the graph is scaled properly
  • Shift the graph horizontally and then vertically to reach its final position
• Example: Transform $f(x) = \cos(x)$ using the following steps: reflect across the x-axis, stretch vertically by a factor of 3, compress horizontally by a factor of 2, shift right by $\pi/4$, and shift up by 1
  1. Reflect: $g(x) = -\cos(x)$
  2. Stretch vertically: $h(x) = -3\cos(x)$
  3. Compress horizontally: $i(x) = -3\cos(2x)$
  4. Shift right: $j(x) = -3\cos(2(x - \pi/4))$
  5. Shift up: $k(x) = -3\cos(2(x - \pi/4)) + 1$

Real-world applications of transformations

• Modeling population growth with exponential functions
  • An exponential function $P(t) = P_0e^{rt}$ can model population growth, where $P_0$ is the initial population, $r$ is the growth rate, and $t$ is time
  • Transformations can represent changes in initial population (vertical shift) or growth rate (horizontal stretch/compression)
  • Example: If a city's population grows 5% annually and has an initial population of 100,000, the transformed function $P(t) = 100,000e^{0.05t}$ models its growth
• Analyzing sound waves with trigonometric functions
  • Sound waves can be modeled using sine or cosine functions, where amplitude represents volume and frequency represents pitch
  • Transformations can simulate changes in volume (vertical stretch/compression) or pitch (horizontal stretch/compression)
  • Example: A sound wave with an amplitude of 2 units and a frequency of 440 Hz can be modeled by $f(t) = 2\sin(880\pi t)$, where $t$ is time in seconds
• Describing motion with quadratic functions
  • Quadratic functions can model the height of an object in motion over time, such as a ball thrown upward
  • Transformations can account for initial height (vertical shift), initial velocity (vertical stretch/compression), and acceleration due to gravity (horizontal stretch/compression)
  • Example: A ball thrown upward from a height of 5 ft with an initial velocity of 20 ft/s can be modeled by $h(t) = -16t^2 + 20t + 5$, where $h$ is height in feet and $t$ is time in seconds