Translations of Functions
Translations involve shifting the graph of a function horizontally or vertically without changing its shape.
Vertical Translation
A vertical translation moves the graph of a function up or down on the coordinate plane. The general form is:
$y = f(x) + b$
where $b$ is the vertical shift.
- If $b > 0$, the graph shifts up by $|b|$ units.
- If $b< 0$, the graph shifts down by $|b|$ units.
Consider the function $f(x) = x^2$. The function $g(x) = x^2 + 3$ is a vertical translation of $f(x)$ by 3 units upward.
Horizontal Translation
A horizontal translation moves the graph of a function left or right on the coordinate plane. The general form is:
$y = f(x - a)$
where $a$ is the horizontal shift.
- If $a > 0$, the graph shifts right by $a$ units.
- If $a< 0$, the graph shifts left by $|a|$ units.
For the function $f(x) = x^2$, the function $h(x) = (x - 2)^2$ is a horizontal translation of $f(x)$ by 2 units to the right.
NoteIt's important to remember that for horizontal translations, the sign inside the parentheses is opposite to the direction of the shift.
Reflections of Functions
Reflections involve flipping the graph of a function over a particular axis.
Reflection in the x-axis
To reflect a function in the x-axis, we negate the function:
$y = -f(x)$
This flips the graph vertically, inverting all y-values.
Reflection in the y-axis
To reflect a function in the y-axis, we negate the input:
$y = f(-x)$
This flips the graph horizontally, reversing all x-values.
ExampleFor $f(x) = x^3$:
- $g(x) = -x^3$ is the reflection of $f(x)$ in the x-axis.
- $h(x) = (-x)^3$ is the reflection of $f(x)$ in the y-axis.
Vertical Stretch of a Function
A vertical stretch (or compression) changes the amplitude of a function. The general form is:
$y = pf(x)$
where $p$ is the scale factor.
- If $|p| > 1$, the graph is stretched vertically.
- If $0< |p| < 1$, the graph is compressed vertically.
- If $p < 0$, the graph is also reflected in the x-axis.
For $f(x) = x^2$, the function $g(x) = 3x^2$ is vertically stretched by a factor of 3.
Horizontal Stretch of a Function
A horizontal stretch (or compression) affects the rate of change of a function. The general form is:
$y = f(qx)$
where $1/q$ is the scale factor.
- If $|q|< 1$, the graph is stretched horizontally.
- If $|q| >1$, the graph is compressed horizontally.
- If $q< 0$, the graph is also reflected in the y-axis.
For $f(x) = x^2$, the function $h(x) = (2x)^2$ is horizontally compressed by a factor of 2.
Common MistakeStudents often confuse horizontal stretches with vertical stretches. Remember, for a horizontal stretch, the factor goes inside the function, while for a vertical stretch, it's outside.
Composite Transformations
In practice, multiple transformations are often applied to a function simultaneously. The order in which these transformations are performed can affect the final result.
ExampleConsider the function $g(x) = 2(x - 3)^2 + 4$. This can be broken down into:
- A horizontal translation of 3 units right: $x \rightarrow (x - 3)$
- A vertical stretch by a factor of 2: $y \rightarrow 2y$
- A vertical translation of 4 units up: $y \rightarrow y + 4$
When working with composite transformations, it's crucial to apply them in the correct order, typically from inside to outside in the function notation.
Using $y = x^2$ as a Parent Function
The quadratic function $y = x^2$ is often used as a parent function to demonstrate transformations. Its simple parabolic shape makes changes easy to visualize.
ExampleLet's transform $y = x^2$ step by step:
- $y = 2x^2$ (vertical stretch by factor 2)
- $y = 2(x - 1)^2$ (horizontal translation 1 unit right)
- $y = 2(x - 1)^2 - 3$ (vertical translation 3 units down)
Using Dynamic Graphing Software
Dynamic graphing software, such as Desmos or GeoGebra, is invaluable for exploring function transformations. These tools allow students to:
- Visualize transformations in real-time
- Experiment with different parameters
- Compare multiple transformations simultaneously
When using graphing software, try creating sliders for transformation parameters. This allows for easy manipulation and observation of how changes affect the graph.
Link to Composite Functions
Function transformations are closely related to composite functions (covered in SL2.5). In fact, many transformations can be expressed as compositions.
ExampleThe transformation $y = 2f(x - 3) + 1$ can be written as a composition: $g(x) = h(f(k(x)))$, where:
- $k(x) = x - 3$ (horizontal translation)
- $f$ is the original function
- $h(x) = 2x + 1$ (vertical stretch and translation)
Sketching
Don't worry too much about scale factors—you can't really sketch them to scale no matter how hard you try anyway.
Focus on the general shape first—take a look at the reflections and how they alter the shape of the graph.
Afterwards, then start understanding how the graph is translated—move the shape you've just drawn a little right or a little up to get the graph you want.
