Question
what is a translation that shifts the graph of a function horizontally
Answers
GPT-4o mini
A translation that shifts the graph of a function horizontally involves adding or subtracting a value from the input variable (usually \(x\)) in the function's equation.
The general form of a translation that shifts a function \(f(x)\) horizontally is as follows:
- If you have the function \(f(x)\) and you want to shift it to the right by \(c\) units, the new function will be:
\[
g(x) = f(x - c)
\]
- Conversely, if you want to shift the function to the left by \(c\) units, the new function will be:
\[
g(x) = f(x + c)
\]
Here, \(c\) is a positive constant.
For example, if the original function is \(f(x) = x^2\):
- To shift it to the right by 3 units, you would have \(g(x) = f(x - 3) = (x - 3)^2\).
- To shift it to the left by 3 units, you would have \(g(x) = f(x + 3) = (x + 3)^2\).
The key takeaway is that subtracting from \(x\) shifts the graph to the right, while adding to \(x\) shifts it to the left.
The general form of a translation that shifts a function \(f(x)\) horizontally is as follows:
- If you have the function \(f(x)\) and you want to shift it to the right by \(c\) units, the new function will be:
\[
g(x) = f(x - c)
\]
- Conversely, if you want to shift the function to the left by \(c\) units, the new function will be:
\[
g(x) = f(x + c)
\]
Here, \(c\) is a positive constant.
For example, if the original function is \(f(x) = x^2\):
- To shift it to the right by 3 units, you would have \(g(x) = f(x - 3) = (x - 3)^2\).
- To shift it to the left by 3 units, you would have \(g(x) = f(x + 3) = (x + 3)^2\).
The key takeaway is that subtracting from \(x\) shifts the graph to the right, while adding to \(x\) shifts it to the left.