Graph transformations are a powerful tool to understand how different functions behave and relate to each other. In this article, we will explore how the graph of y=3(x+1)^2 is related to its parent function y=x^2, which is a simple quadratic function. We will also learn how to use some basic rules to perform graph transformations and sketch the graphs of different functions.
Contents
What is a parent function?
A parent function is the simplest form of a function that belongs to a certain family or type of functions. For example, the parent function of all linear functions is y=x, and the parent function of all exponential functions is y=a^x, where a is a constant. Parent functions have some basic properties and shapes that can help us identify and graph other functions in the same family.
The parent function of all quadratic functions is y=x^2, which has a U-shaped curve called a parabola. The graph of y=x^2 is shown below:
!graph of y=x^2)
What are graph transformations?
Graph transformations are operations that change the position, shape, or size of a graph by applying some rules to the original function. There are four main types of graph transformations:
- Translations: These move the graph up, down, left, or right by adding or subtracting constants to the function.
- Reflections: These flip the graph over an axis by multiplying the function by -1.
- Stretches: These make the graph taller or wider by multiplying the function by a constant greater than 1.
- Compressions: These make the graph shorter or narrower by multiplying the function by a constant between 0 and 1.
We can use these transformations to create new functions from parent functions and compare their graphs.
The function y=3(x+1)^2 is a quadratic function that is derived from the parent function y=x^2 by applying two transformations: a horizontal translation and a vertical stretch. Let’s see how these transformations affect the graph of y=x^2.
Horizontal translation
A horizontal translation moves the graph left or right by adding or subtracting a constant to the x-value in the function. For example, if we add 2 to x, we get y=(x+2)^2, which moves the graph 2 units to the left. If we subtract 3 from x, we get y=(x-3)^2, which moves the graph 3 units to the right.
The rule for horizontal translations is:
- If we add C to x, we move the graph C units to the left.
- If we subtract C from x, we move the graph C units to the right.
In our case, we have y=3(x+1)^2, which means we add 1 to x. Therefore, we move the graph 1 unit to the left. The vertex of the parabola shifts from (0,0) to (-1,0), as shown below:
!graph of y=3(x+1)^2 with horizontal translation”))
Vertical stretch
A vertical stretch makes the graph taller by multiplying the whole function by a constant greater than 1. For example, if we multiply y=x^2 by 2, we get y=2x^2, which doubles the height of the graph. If we multiply y=x^2 by 4, we get y=4x^2, which quadruples the height of the graph.
The rule for vertical stretches is:
- If we multiply the function by C, where C > 1, we stretch the graph by a factor of C in the y-direction.
In our case, we have y=3(x+1)^2, which means we multiply y=x^2 by 3. Therefore, we stretch the graph by a factor of 3 in the y-direction. The vertex of the parabola remains at (-1,0), but its width becomes narrower and its maximum value becomes higher, as shown below:
!graph of y=3(x+1)^2 with vertical stretch”))
Conclusion
We have seen how the graph of y=3(x+1)^2 is related to its parent function y=x^2 by applying two transformations: a horizontal translation and a vertical stretch. We have also learned how to use some basic rules to perform these transformations and sketch the graphs of different functions.
Graph transformations are useful for understanding how different functions behave and relate to each other. They can also help us model real-world situations and solve problems involving functions. We hope this article has helped you learn more about graph transformations and how to apply them to quadratic functions.
