IMG_2938.jpeg

# Chapter 7 - Transformations of Functions ## 7.1 Horizontal and Vertical Shifts ### Vertical Shift Given a function $f(x)$, if we define a new function as $g(x) = f(x) + k$, where $k$ is a constant, then $g(x)$ is a vertical shift of the function $f(x)$, where all the output values of $f(x)$ are...

# Chapter 7 - Transformations of Functions ## 7.1 Horizontal and Vertical Shifts ### Vertical Shift Given a function $f(x)$, if we define a new function as $g(x) = f(x) + k$, where $k$ is a constant, then $g(x)$ is a vertical shift of the function $f(x)$, where all the output values of $f(x)$ are increased by $k$. - If $k$ is positive, the graph will shift **up** - If $k$ is negative, the graph will shift **down** ### Horizontal Shift Given a function $f(x)$, if we define a new function as $g(x) = f(x - h)$, where $h$ is a constant, then $g(x)$ is a horizontal shift of the function $f(x)$. - If $h$ is positive, the graph will shift to the **right** - If $h$ is negative, the graph will shift to the **left** ## 7.2 Vertical Stretch and Compression Given a function $f(x)$, if we define a new function $g(x) = af(x)$, where $a$ is a constant, then $g(x)$ is a vertical stretch or vertical compression of the function $f(x)$. - If $a > 1$, then the graph will be stretched vertically - If $0 < a < 1$, then the graph will be compressed vertically - If $a < 0$, then there will be combination of vertical stretch or compression with a vertical reflection ## 7.3 Horizontal Stretch and Compression Given a function $f(x)$, if we define a new function $g(x) = f(bx)$, where $b$ is a constant, then $g(x)$ is a horizontal stretch or horizontal compression of the function $f(x)$. - If $b > 1$, then the graph will be compressed horizontally - If $0 < b < 1$, then the graph will be stretched horizontally - If $b < 0$, then there will be a combination of horizontal stretch or compression with a horizontal reflection ## 7.4 Combining Transformations ### Vertical Shift - Outside changes affect the output (vertical) of the function, shifting the transformed function up or down. ### Horizontal Shift - Inside changes affect the input (horizontal) of the function, shifting the transformed function left or right. ### Vertical Stretch and Compression - Outside multiplication stretches or compresses the graph vertically. ### Horizontal Stretch and Compression - Inside multiplication stretches or compresses the graph horizontally. ### Vertical and Horizontal Reflections - When combining transformations, it is very important to consider the order of the transformations. For example, vertically shifting by $k$ then vertically stretching by $a$ does not produce the same graph as vertically stretching by $a$ then vertically shifting by $k$. ## 7.5 Domain and Range ### Domain The **domain** of a function is the set of all possible input values for the function. ### Range The **range** of a function is the set of all possible output values for the function. ## 7.6 Inverse Functions Given a function $f(x)$, we can define an **inverse** function $f^{-1}(x)$, which is a function that undoes $f(x)$. In other words, $f^{-1}(f(x)) = x$. ### Finding an Inverse Function 1. Replace $f(x)$ with $y$ 2. Swap $x$ and $y$ 3. Solve for $y$ 4. Replace $y$ with $f^{-1}(x)$ ### One-to-One Function A **one-to-one function** is a function where each input value corresponds to exactly one output value, and each output value corresponds to exactly one input value. Only one-to-one functions have inverse functions.