Horizontal Translation Direction and Parameter Sign
Horizontal translation direction is counterintuitive: in f(x) = a^(x-h), a positive h shifts the graph right, while a negative h shifts it left — opposite to what the sign suggests. Grade 11 students in enVision Algebra 1 (Chapter 6: Exponents and Exponential Functions) learn this rule by finding the x-value that makes the exponent zero. For x − 3 = 0, the reference point is x = 3 (shift right 3 units); for x + 2 = 0, it is x = −2 (shift left 2 units). The memory key: subtract means right, add means left.
Key Concepts
For horizontal translations $f(x) = a^{(x h)}$, the direction of translation is opposite to the sign of $h$: when $h 0$, the graph shifts right by $h$ units; when $h < 0$, the graph shifts left by $|h|$ units.
Common Questions
Which direction does f(x) = a^(x−h) shift when h is positive?
A positive h shifts the graph to the right by h units, even though the sign inside the exponent is negative.
Which direction does f(x) = a^(x+h) shift when h is positive?
Writing x + h is equivalent to x − (−h), so the graph shifts to the left by h units.
Why does horizontal translation direction seem counterintuitive?
Because subtracting h from x (written as x − h) moves the graph right, while adding h (written as x + h) moves it left — opposite to the sign.
How do you determine the horizontal shift amount from the equation?
Set the expression in the exponent equal to zero and solve for x. That x-value is the amount and direction of the horizontal shift.
What is the horizontal shift of f(x) = 2^(x−3)?
Setting x − 3 = 0 gives x = 3, so the graph shifts 3 units to the right.
What is the horizontal shift of f(x) = 3^(x+4)?
Setting x + 4 = 0 gives x = −4, so the graph shifts 4 units to the left.