What is a Horizontal Shift of a Function?
A horizontal shift adds or subtracts a constant to or from every x-value, leaving the y-coordinate unchanged. The basic rules for shifting a function along a horizontal (x) are:
Compared to a base graph of f(x),
- y = f(x + h) shifts h units to the left,
- y = f(x – h) shifts h units to the right,
Where h > 0.
Look carefully at what the positive or negative added value h is doing: it’s the opposite of what you might expect. Positive values of h shift in the negative direction along the number line and negative h values shift the the positive direction.
Example of a Horizontal Shift
A horizontal shift of the function f(x) = x2 of 2 units (i.e. h = 2) results in:
- f(x) = x2 + 3 (3 units to the left),
- f(x) = x2 + 3 (3 units to the right)
The following graph shows the base function f(x) = x2 and the two “new” graphs created when we added 2 or subtracted 2.
Example question #1: How are the graphs of y = √(x) and y = √(x + 1) related?
Step 1: Compare the right sides of both equations and note any differences:
- √(x + 1)
The difference between the equations is a “+ 1”.
Step 2: Choose a rule based on whether Step 1 was positive or negative:
Step 1 for this example was positive (+ 1), so that’s rule 1:
y = f(x + h) shifts h units to the left
Step 3: Place your base function (from the question) into the rule, in place of “x”:
y = f(√(x) + h) shifts h units to the left
Step 4: Place “h” — the difference you found in Step 1 — into the rule from Step 3:
y = f(√(x) + 2) shifts 2 units to the left
Example question #2: The following graph shows how the average cost of a new car tire compares from Jacksonville, Florida (red) to Miami, Florida (Blue). Write a formula for the transformation of g (blue graph) to f (red graph).
Solution: The graph g shifts 10 units to the right of f, so:
g(x) = f(x – 20) .
If you’re not sure about how I arrived at this formula, the following few steps break it down into simple parts:
Step 1: Decide which direction the graph is traveling (left, or right?). The question asks is for a formula for the transformation of g (blue graph) as a transformation of f (red graph). In other words, what direction do we need to travel to turn f into g? A look at the graph shows that moves to the right.
Step 2: Take your answer from Step 1 and then refer to the rules to tell you whether it’s a positive or negative shift.
Rule 2 states:
y = f(x – h) shifts h units to the right.
That means moving to the right must mean we have a “-” shift.
Put this value aside for a moment.
Step 3: Locate two x-values on the horizontal axis: one for each graph:
- g (blue graph) = 70
- f (red graph) = 90.
Step 4: Subtract the lowest number in Step 3 from the highest:
90 – 70 = 20.
Step 5: Combine your answers from Steps 2 and 4:
Step 6: Place your answer from Step 5 into the rule you chose in Step 2, replacing the “h” with your value (- 20 in this example):
g(x) = f(x – 20)
Don’t forget to rename the formula with the one given in the question!
Graph created with Desmos.com.
Stephanie Glen. "Horizontal Shift of a Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/horizontal-shift-of-a-function/
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