Creating Comparison Tables for Function Transformations
Creating comparison tables for function transformations is a Grade 11 Algebra 1 technique from enVision Chapter 3 that reveals transformation patterns by computing f(x) and g(x) side-by-side for the same inputs. For f(x) = 2x + 1 and g(x) = 2x + 4, a table at x = 0, 1, 2 shows outputs (1,4), (3,6), (5,8) — each g(x) value is exactly 3 more than f(x), confirming a vertical shift of 3. Using convenient x-values like -2, -1, 0, 1, 2 makes arithmetic easier and reveals whether outputs shift vertically, scale, or shift horizontally.
Key Concepts
To compare an original function $f(x)$ with its transformation $g(x)$, create a table with columns for $x$, $f(x)$, and $g(x)$ using the same input values. Calculate outputs systematically to identify patterns in how the transformation affects each point.
Common Questions
What is the purpose of a comparison table for transformations?
It places original and transformed function outputs side-by-side for the same inputs, making the transformation pattern visible through the differences or ratios in outputs.
How does a comparison table confirm a vertical shift?
If g(x) - f(x) is constant for all x-values tested, the transformation is a vertical shift by that constant amount.
For f(x) = 2x + 1 and g(x) = 2x + 4, what does the comparison table show?
At x=0: f=1, g=4. At x=1: f=3, g=6. At x=2: f=5, g=8. The difference is always 3, confirming a vertical shift up by 3.
Which x-values are best to use in a comparison table?
Simple integers like -2, -1, 0, 1, 2 minimize arithmetic errors and often produce patterns that are easy to recognize.
How can a table distinguish a horizontal from a vertical shift?
For a horizontal shift, the outputs of g(x) match outputs of f(x) but at different input values. For a vertical shift, outputs differ by a constant at every matching input.
What pattern in outputs indicates a vertical stretch rather than a shift?
If g(x) = k*f(x), the ratio g(x)/f(x) is constant, not the difference. This signals a vertical stretch or compression by factor k.