Identifying Translations from Tables
Identifying translations from tables involves comparing y-values between an original exponential function and its translated version. In Grade 11 enVision Algebra 1 (Chapter 6: Exponents and Exponential Functions), for vertical translations, students subtract f(x) from g(x) at the same x-value to find k = g(x) − f(x). For horizontal translations, they find x-values where both functions produce the same output and calculate h = x₁ − x₂. This table-based approach makes translations concrete and connects numeric patterns to algebraic transformation parameters.
Key Concepts
To identify vertical translations: Compare $y$ values for the same $x$ value between $f(x) = a^x$ and $g(x) = a^x + k$, where $k = g(x) f(x)$.
To identify horizontal translations: Find $x$ values that produce the same $y$ value in both functions, where $h = x 1 x 2$ for $f(x 1) = g(x 2)$.
Common Questions
How do you identify a vertical translation from a table of values?
Compare g(x) − f(x) at any matching x-value. If this difference is constant across all x-values, that constant is the vertical shift k.
How do you identify a horizontal translation from a table of values?
Find an x-value in f(x) and an x-value in g(x) that produce the same output. The difference between those x-values is the horizontal shift h.
What does a constant column difference in a table indicate?
If g(x) − f(x) is the same value for every row in the table, the function has been vertically shifted by that constant.
If f(2) = 4 and g(5) = 4, what is the horizontal shift?
h = 5 − 2 = 3. The function g was shifted 3 units to the right compared to f.
Can you identify both translations simultaneously from a table?
Yes, but it requires more careful analysis — check first whether outputs shift by a constant (vertical), then examine input positions of equal outputs (horizontal).
Why does table analysis help understand function translations?
Tables make abstract algebraic transformations concrete by showing exactly how input-output pairs change numerically when a function is translated.