6.4 Comparing and Combining Functions Across Representations
Key Takeaways
- Standard AII-F.IF.9 asks students to compare properties of two functions presented in different ways - algebraically, graphically, numerically in a table, or by a verbal description - and NYSED extends the families to include square root and cube root functions alongside polynomial, exponential, logarithmic, and trigonometric ones.
- The reliable protocol is to name the property, extract that one property from each representation using the method appropriate to that representation, and only then compare; never convert every representation into an equation first.
- A table of equally spaced inputs identifies its family through differences and ratios: constant first differences signal linear, constant second differences signal quadratic, and a constant ratio signals exponential.
- Standard AII-F.BF.1b requires combining standard function types using arithmetic operations, so a profit model is a revenue function minus a cost function and inherits the domain restrictions of both.
- When two functions are compared over an interval rather than at a point, the fair comparison statistic is the average rate of change [f(b) - f(a)]/(b - a) evaluated on the same interval for both.
6.4 Comparing and Combining Functions Across Representations
[!NOTE] Regents Standard Alignment: AII-F.IF.9 - compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). NYSED's Algebra II note widens the families: "Tasks also involve square root and cube root functions, in addition to polynomial, exponential, logarithmic and trigonometric functions." AII-F.BF.1b - combine standard function types using arithmetic operations.
Every F-IF.9 item has the same shape. You are handed two functions in two different formats and asked which one has the greater something. The format asymmetry is the difficulty; the underlying comparison is usually easy once each property is extracted.
1. The Extraction Table
Do not convert everything into an equation. Pull the one property you need directly from whatever form you were given.
| Property | From an Equation | From a Graph | From a Table | From a Verbal Description |
|---|---|---|---|---|
| $y$-intercept | Evaluate $f(0)$ | Read where the curve crosses the vertical axis | Read the row where $x = 0$ | The stated "initial", "starting", or "at time zero" value |
| Zeros | Solve $f(x) = 0$ | Read the horizontal-axis crossings | Find rows where the output is $0$ or where the sign changes | Stated "breaks even", "hits the ground", "runs out" |
| Maximum or minimum | Vertex $x = -\frac{b}{2a}$ for a quadratic; otherwise 2nd TRACE -> max/min | Read the peak or valley | Find the largest or smallest listed output, checking symmetry around it | Stated "peaks at", "the most it ever reaches" |
| Average rate of change on $[a,b]$ | $\dfrac{f(b) - f(a)}{b - a}$ | Slope of the segment joining the two points | Same quotient using two table rows | Stated "increases by ... per ..." |
| End behavior | Leading term for polynomials; base for exponentials | Follow the arms | Extend the trend of the last rows with caution | Stated "levels off", "grows without bound" |
| Period / amplitude / midline | $P = \frac{2\pi}{ | B | }$, $ | A |
Fingerprinting a Table
For equally spaced inputs:
- Constant first differences $\Rightarrow$ linear.
- Constant second differences $\Rightarrow$ quadratic.
- Constant ratio between consecutive outputs $\Rightarrow$ exponential.
- Neither pattern, but a curve that flattens as $x$ grows and is defined only for $x \geq$ some value $\Rightarrow$ suspect a square root; defined everywhere with an inflection $\Rightarrow$ suspect a cube root.
2. Worked Comparison 1: Equation Versus Table
Problem: Function $f$ is defined by $f(x) = 2x^2 - 8x + 3$. Function $g$ is given by the table below. Which function has the greater minimum value, and by how much?
| $x$ | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| $g(x)$ | 6 | 0 | $-2$ | 0 | 6 |
- Confirm the family of $g$. First differences: $-6, -2, 2, 6$. Second differences: $4, 4, 4$. Constant second differences confirm a quadratic.
- Minimum of $g$. The table is symmetric about $x = 2$, where the output $-2$ is the smallest listed value and the outputs rise identically on both sides. The minimum value of $g$ is $-2$.
- Minimum of $f$. For a quadratic in standard form the vertex sits at $x = -\dfrac{b}{2a} = -\dfrac{-8}{2(2)} = 2$. Then $f(2) = 2(4) - 8(2) + 3 = 8 - 16 + 3 = -5$. Because $a = 2 > 0$, this is a minimum.
- Compare. $-2 > -5$, so $g$ has the greater minimum, by 3 units.
[!WARNING] Both functions bottom out at $x = 2$, which makes it tempting to say they are "the same." The question asks about the minimum value - an output - not its location. Confusing the two is the classic F-IF.9 deduction.
3. Worked Comparison 2: Verbal Versus Graph, Using Average Rate of Change
Problem: A biologist describes colony $A$ verbally: it starts at 40 cells and triples every hour. Colony $B$ is shown on a graph that passes through $(0, 400)$, $(1, 600)$, $(2, 800)$, and $(3, 1000)$. Over the interval from $t = 0$ to $t = 3$ hours, which colony has the greater average rate of change?
- Model $A$ from the description. Tripling per hour is exponential: $A(t) = 40(3)^t$. Then $A(0) = 40$ and $A(3) = 40(27) = 1080$.
- Read $B$ from the graph. The listed points rise by a constant 200 per hour, so $B$ is linear with $B(0) = 400$ and $B(3) = 1000$.
- Compare. $346.7 > 200$, so colony $A$ has the greater average rate of change over $[0, 3]$.
- Interpret honestly. Colony $B$ is far larger for the first two hours - $A(1) = 120$ against $B(1) = 600$ - and only overtakes $B$ between $t = 2$ ($A = 360$, $B = 800$) and $t = 3$ ($A = 1080$, $B = 1000$). "Greater average rate of change on an interval" is not the same claim as "larger throughout."
4. Combining Functions with Arithmetic Operations (AII-F.BF.1b)
Standard AII-F.BF.1b asks you to build a new function from two standard types using $+$, $-$, $\times$, or $\div$. The workhorse example in context is profit.
Worked Combination
Problem: A small manufacturer's weekly revenue in dollars from selling $x$ units is $R(x) = 120x - 2x^2$, and weekly cost is $C(x) = 300 + 20x$. Write the profit function, determine the profit at 10 units, and find the production level that maximizes profit.
- Combine. Distribute the subtraction across both terms of $C(x)$ - dropping the sign on $+20x$ is the single most common error here.
- Evaluate at 10 units. Weekly profit at 10 units is $$500$.
- Maximize. The profit function is a downward-opening parabola, so its maximum is at the vertex: Maximum weekly profit is $$950$ at 25 units.
Rules a Combined Function Inherits
| Operation | Resulting Domain | Watch For |
|---|---|---|
| $(f + g)(x)$, $(f - g)(x)$, $(f \cdot g)(x)$ | The intersection of the domains of $f$ and $g$ | A radical or logarithmic piece narrows the whole combined domain |
| $\left(\dfrac{f}{g}\right)(x)$ | The intersection, minus every $x$ with $g(x) = 0$ | Zeros of the denominator function are excluded even if $f$ is fine there |
For example, if $f(x) = \sqrt{x - 1}$ and $g(x) = x - 4$, then $(f + g)(x) = \sqrt{x - 1} + x - 4$ has domain $[1, \infty)$, while $\left(\frac{f}{g}\right)(x)$ has domain $[1, 4) \cup (4, \infty)$.
Function p is defined by p(x) = 3x² - 12x + 4. Function q is given by the table with q(0) = 7, q(1) = 4, q(2) = 3, q(3) = 4, and q(4) = 7. Which statement correctly compares the minimum values of the two functions?
Colony A starts with 40 cells and triples every hour. Colony B is graphed through the points (0, 400), (1, 600), (2, 800), and (3, 1000). Which statement about the interval from t = 0 to t = 3 hours is correct?
A shop has revenue R(x) = 120x - 2x² dollars and cost C(x) = 300 + 20x dollars for x units sold per week. Which expression gives the weekly profit, and what is the profit at 10 units?