Horizontal Translations of Exponential Functions
Grade 9 students in California Reveal Math Algebra 1 learn how horizontal translations shift the graph of an exponential function left or right without changing its shape. Starting from parent function f(x)=b^x, a horizontal translation produces g(x)=b^(x-h). When h>0 the graph shifts right by h units, and when h<0 it shifts left by |h| units. For example, g(x)=2^(x-3) shifts right 3 units and g(x)=3^(x+4) shifts left 4 units. Critically, the horizontal asymptote stays at y=0 and the domain and range are unchanged, but the y-intercept does change because the graph has moved horizontally.
Key Concepts
A horizontal translation of an exponential function shifts the graph left or right. Starting from the parent function $f(x) = b^x$, a horizontal translation produces:.
$$g(x) = b^{(x h)}$$.
Common Questions
How does a horizontal translation affect an exponential function?
A horizontal translation shifts every point on the parent graph f(x)=b^x left or right without changing its shape. The value h in g(x)=b^(x-h) controls the direction: positive h shifts right, negative h shifts left.
Does horizontal translation change the horizontal asymptote?
No. The horizontal asymptote stays at y=0 for a horizontal translation. The domain and range also remain unchanged.
Does horizontal translation change the y-intercept?
Yes. Unlike vertical translations, a horizontal shift does change the y-intercept because the graph has moved sideways. For example, g(x)=(1/2)^(x-1) has y-intercept (0, 1/2) rather than (0,1).
How do you identify the direction of a horizontal shift from the equation?
Look at g(x)=b^(x-h). Subtracting a positive h shifts right; subtracting a negative h (which is the same as adding) shifts left. g(x)=3^(x+4) rewrites as 3^(x-(-4)), so h=-4 means a leftward shift.
Give an example of a horizontal translation of an exponential function.
g(x)=2^(x-3) is the parent f(x)=2^x shifted 3 units to the right. The asymptote remains y=0, but the graph's position has moved rightward.
Which unit and textbook covers horizontal translations of exponential functions?
This skill is from Unit 8: Exponential Functions in California Reveal Math Algebra 1, Grade 9.