Compare functions across representations

Try

A function can arrive as a graph, a table, a formula or a description. To compare two functions, choose the same question and the same domain: value at a particular input, largest output, sign, or average rate. Saying one function is larger is incomplete unless it means at one input, everywhere, or at its maximum.

Predict

Predict whether the function with the larger maximum must be larger at every input. It need not. Predict whether the largest entry in a sampled table is the function’s true maximum. A curve may rise higher between observations. These two mistakes come from changing the comparison question without noticing.

Experiment

Supplied Graph A is a complete quadratic model on 0≤x≤4, not merely five observations. Its coordinate table provides equivalent accessible landmarks.

x	A(x)
0	2
1	5
2	6
3	5
4	2

The curved graph, its stated quadratic form and domain determine the model between landmarks. Compare it with B(x)=7−2(x−1)² on the same domain. Separately, treat the three points (0,1),(2,7),(4,3) as observations only: no interpolation has been specified for that third function.

Understand

A graph’s horizontal coordinate is the input and its vertical coordinate is the output. Read scale labels before interpreting a point. An intercept is where the graph meets an axis; a maximum is the greatest attained output on the stated domain. A function can have a maximum over a whole interval, not only at one point. Increasing and decreasing describe changes as input increases; positive and negative describe the sign of the output. A negative balance can be increasing while remaining below zero.

To compare Graph A with B at a chosen input, read A’s plotted height and substitute that same input into B. To compare maxima, locate each peak and check that its input lies in the shared domain. To compare average rates, use the same pair of inputs and divide each output change by the same input change. These are different comparisons and may produce different winners.

Quadratic forms reveal different features. A factored rule h(t)=t(6−t) shows zeros at 0 and 6. Completing the square gives h(t)=9−(t−3)², showing a maximum of 9 at t=3. Both expressions define the same algebraic parabola, but a height model may restrict the input to a particular time interval. The algebraic expression does not authorize extending a physical scenario beyond its stated domain.

A sampled table requires special care. A maximum sampled output is only a lower bound on the maximum of any full model containing those points. Additional assumptions such as linear interpolation, monotonicity or a specified quadratic family can change what is inferable. State those assumptions rather than silently connecting dots and treating the drawn curve as observed evidence.

Apply

Build a graph from a verbal account with a negative starting balance, a rising segment, a plateau and a falling segment. A written coordinate description is acceptable when it specifies exact vertices, how they are joined, endpoint inclusion, axes, units and domain. Its rubric must assess those features; a single numerical maximum cannot certify the whole construction.

Continue

When representations differ, translate only what is needed for the comparison and keep the domain visible. The next lessons add missing graph families so that a corner, a root curve and a whole-unit step can be recognized and constructed rather than treated as interchangeable shapes.

Continue to Construct cube-root and absolute-value graphs.