Comparing Key Features of Two Functions
Comparing key features of two functions is a Grade 9 Algebra 1 skill in California Reveal Math (Unit 2: Relations and Functions). Students build a side-by-side table listing y-intercepts, x-intercepts, relative maxima and minima, intervals of increase and decrease, and end behavior for each function. For example, comparing f(x) = -x^2 + 6x and g(x) = -2x^2 + 8x shows f peaks at (3, 9) while g peaks at (2, 8), and f stays positive longer on (0, 6) versus g on (0, 4).
Key Concepts
To compare two functions $f(x)$ and $g(x)$, build a side by side table of their key features:.
$$\begin{array}{|l|c|c|} \hline \textbf{Feature} & f(x) & g(x) \\ \hline \text{y intercept} & f(0) & g(0) \\ \text{x intercept(s)} & f(x)=0 & g(x)=0 \\ \text{Relative max/min} & (x 1,\, f(x 1)) & (x 2,\, g(x 2)) \\ \text{Increasing on} & (a,\, b) & (c,\, d) \\ \text{Decreasing on} & (b,\, e) & (d,\, h) \\ \text{End behavior} & \text{describe} & \text{describe} \\ \hline \end{array}$$.
Common Questions
What features should you compare when analyzing two functions side by side?
Compare y-intercepts, x-intercepts, relative maxima and minima with their coordinates, intervals of increase and decrease, and end behavior. Build a structured table so differences stand out row by row.
How do you compare f(x) = -x^2 + 6x and g(x) = -2x^2 + 8x?
f has relative maximum of 9 at x=3; g has relative maximum of 8 at x=2. f stays positive on (0,6); g on (0,4). Both have the same y-intercept of 0 and x-intercepts at 0.
Why use a table format to compare two functions?
A table prevents the common error of focusing on only one feature while missing others. It forces systematic comparison so you can answer questions like which function peaks higher or reaches zero sooner.
How do comparison tables apply to real-world problems?
In projectile motion, comparing maxima tells you which ball reaches a greater height. In population growth, comparing rates of change tells you which city grows faster after a given year.
What is end behavior and why does it matter when comparing functions?
End behavior describes what happens to function values as x approaches positive or negative infinity. It determines which function eventually dominates for large x, important in revenue or growth comparisons.