Why does shifting a function to the right require a smaller input variable?

In summary, when shifting a graph horizontally, you are changing the input variable before performing the function. This means that for a positive slope, you need to use a constant that is smaller than the current input in order to shift the graph to the right. Shifting a graph up or down simply involves adding or subtracting a constant to the result of the function.
  • #1
greenneub
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0

Homework Statement


Could someone please explain, in very simple words, the horizontal shift of a graph? I've used the search button, textbooks, and google and I'm just not comprehending this. Why does the graph f(x-1) shift f(x) 1 unit to the right and not left?


Homework Equations



f(x) = x Let's keep it simple.

The Attempt at a Solution


If I take some ordered pairs, say (0,0), (1, 1), (2,2) as points on f(x), then try to create the new ordered pairs with f(x-1), shouldn't they be (-1, 0), (0, 1) (1,2) and the f(x-1) graph be moved 1 unit to the left?
 
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  • #2
Because the first thing you do in evaluating f(x- 1) is subtract 1 from x. In other words, if g(x)= f(x-1), g(1)= f(0), g(2)= f(1), etc. The point on the original graph that was at 0 is not at 1, that was at 1 is now at 2, etc.
 
  • #3
Take the graph of y = x^2. It's vertex is at the origin. Now the graph of (x-1)^2 is shifted to the right because to make y = 0, you have to add 1 etc.
 
  • #4
When you shift a function up or down, you have already performed the function f(x) for all the points and you are simply moving the curve up or down the y-axis by adding or subtracting a constant to the result.

When shifting a function left or right, you are shifting the input variable x prior to performing the function. So, for example, if the curve has a positive slope and you are trying to shift it right, then the input variable needs to be some constant SMALLER than the current input so that the resulting curve shifts to the right. If you made the input variable larger, you would have a result for each point that is further up and out on the curve, resulting in a shift of the curve to the left, which is the opposite of what you wanted.
 

1. What is a horizontal shift of function?

A horizontal shift of function is a transformation that moves a function horizontally along the x-axis without changing its shape. It is also known as a translation.

2. How is a horizontal shift of function represented in an equation?

A horizontal shift of function is represented by adding or subtracting a value from the input (x) in the function. For example, f(x + 2) represents a shift of 2 units to the left, while f(x - 3) represents a shift of 3 units to the right.

3. What is the difference between a positive and negative horizontal shift of function?

A positive horizontal shift moves the function to the right, while a negative horizontal shift moves the function to the left. This is because a positive value added to the input (x) will result in a larger x-value, while a negative value will result in a smaller x-value.

4. How does a horizontal shift affect the graph of a function?

A horizontal shift changes the position of the graph of a function without changing its shape. The original graph will shift either to the left or right, depending on the value added or subtracted from the input (x).

5. Can a horizontal shift of function be combined with other transformations?

Yes, a horizontal shift of function can be combined with other transformations, such as vertical shifts, reflections, and stretches/compressions. These transformations are applied in a specific order, known as the transformation rule, to determine the final position of the function's graph.

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