In the world of mathematics, understanding how functions transform is crucial. These transformations provide insights into the behaviour of functions and their graphical representations. In this section, we will explore the basic transformations of functions, focusing on translations, reflections, and dilations.
Translations
A translation is a shift of a function either vertically, horizontally, or both. It doesn't alter the shape of the function but merely changes its position on the coordinate plane.
Horizontal Translations
- Definition: A horizontal translation shifts the graph of a function to the left or right.
- Mathematical Representation: If f(x) is translated h units to the right, the new function is represented as f(x - h).
- Example: The function y = x2 can be translated 3 units to the right to get y = (x - 3)2.This means every point on y = x2 moves 3 units to the right to form the graph of y = (x - 3)2.
Vertical Translations
- Definition: A vertical translation shifts the graph of a function upwards or downwards.
- Mathematical Representation: If f(x) is translated k units upwards, the new function is f(x) + k.
- Example: The function y = x2 can be translated 4 units up to get y = x2 + 4.This means every point on y = x2 moves 4 units up to form the graph of y = x2 + 4.
Reflections
Reflections create a mirror image of a function over a specific axis. The shape remains the same, but the orientation changes.
Reflection over the x-axis
- Definition: This type of reflection flips the function upside down.
- Mathematical Representation: The reflection of f(x) over the x-axis is given by -f(x).
- Example: The reflection of y = x2 over the x-axis is y = -x2.This means the parabola, which opens upwards for y = x2, will now open downwards for y = -x2.
Reflection over the y-axis
- Definition: This reflection creates a mirror image of the function over the y-axis.
- Mathematical Representation: The reflection of f(x) over the y-axis is f(-x).
- Example: The reflection of y = x3 over the y-axis is y = (-x)3.This means the cubic curve of y = x3 will be flipped horizontally to form the curve of y = (-x)3.
Dilations
Dilations either stretch or compress a function. They can change the shape of the function but maintain its basic form.
Vertical Dilation
- Definition: A vertical dilation either stretches or compresses the function away from or towards the x-axis.
- Mathematical Representation: A function f(x) stretched vertically by a factor of a is represented as af(x).
- Example: Stretching y = x2 by a factor of 2 vertically gives y = 2x2.This means the parabola will be narrower than y = x2.
Horizontal Dilation
- Definition: A horizontal dilation either stretches or compresses the function away from or towards the y-axis.
- Mathematical Representation: A function f(x) compressed horizontally by a factor of b is f(bx).
- Example: Compressing y = x3 by a factor of 2 horizontally results in y = (2x)3.This means the cubic curve will be horizontally compressed, making it steeper.
Example Questions
1. Question: Given the function y = x2, describe the transformation for y = (x - 4)2 + 5.
- Answer: The function is translated 4 units to the right and 5 units up.
2. Question: What is the reflection of y = x2 over the y-axis?
- Answer: The reflection is y = (-x)2, which remains y = x2 since squaring a negative number gives a positive result.
3. Question: If y = x3 is compressed by a factor of 3 horizontally, what is the new function?
- Answer: The new function is y = (3x)3 =27x3 .
4. Question: How would you represent a function y = f(x) that has been stretched vertically by a factor of 3 and translated 2 units down?
- Answer: The transformed function would be y = 3f(x) - 2.
FAQ
Transformations can change the intercepts of a function. A vertical translation will shift the y-intercepts up or down, while a horizontal translation will move the x-intercepts left or right. Reflections over the x-axis or y-axis can change the quadrant in which the intercepts lie, potentially moving them to the opposite side of the axis. Dilations can either bring the intercepts closer to or farther from the origin, depending on whether the dilation is a compression or a stretch. It's essential to consider these effects when analysing the transformed function's graph.
Transformations can change the appearance of trigonometric functions but not their inherent periodic nature. For instance, translating or reflecting a sine or cosine function will move or flip its graph, but the function will still repeat its values in regular intervals. However, horizontal dilations can change the length of one period. For example, if the sine function is horizontally dilated by a factor of 2, its period will be halved. It's crucial to recognise the effect of transformations on the periodicity to understand the function's behaviour over different intervals.
Yes, translations and reflections don't change the shape of a function's graph. While they can move the graph to a different location on the coordinate plane or flip it over an axis, the inherent shape remains the same. For example, translating a parabola upwards will shift its position, but it will still look like a parabola. Similarly, reflecting a cubic function over the y-axis will produce a mirror image, but the curve's nature remains unchanged. Only dilations and some advanced transformations can alter the original shape of a function's graph.
Transformations like translations and vertical dilations won't change the overall increasing or decreasing nature of a function. However, reflections can. Reflecting a function over the x-axis will change its direction. For instance, an increasing function will become decreasing after such a reflection. Similarly, reflecting over the y-axis can reverse the direction in which the function increases or decreases. It's vital to understand these effects, especially when analysing real-world scenarios modelled by functions, as the behaviour of the function can provide insights into the situation it represents.
When multiple transformations are applied to a function, they are performed in a specific order. For instance, if a function is first reflected over the x-axis and then translated upwards, the reflection is done first, followed by the translation. It's essential to understand the sequence of transformations to accurately determine the final position and shape of the function. However, some transformations, like translations, are commutative, meaning their order doesn't affect the outcome. But, for non-commutative transformations like dilation followed by a reflection, the order matters and can produce different results.
Practice Questions
The function y = (x + 3)2 - 2 represents a horizontal translation of 3 units to the left and a vertical translation of 2 units down from the original function y = x2. To sketch the graph, start with the standard parabola of y = x2 and shift every point 3 units to the left and 2 units down. The vertex of the new parabola will be at the point (-3, -2).
If the point P(a, b) lies on f(x), after reflecting over the x-axis, its y-coordinate will become the negative of its original value, making the point P'(a, -b). After translating 4 units upwards, the y-coordinate of P' will increase by 4 units, resulting in the point P''(a, -b + 4) or P''(a, 4 - b). Thus, the coordinates of the point when it lies on g(x) will be (a, 4 - b).
