Compare key features (intercepts, rate of change, maximums, and minimums) of two functions each represented differently, such as one as an equation and one as a table or graph (MA.912.F.1.5).

What this topic is asking

MA.912.F.1.5 asks you to compare two functions given in different forms, for example one as an equation and one as a table or graph. The B.E.S.T. Algebra 1 EOC tests whether you can pull the same feature (a slope, a $y$-intercept, a maximum) out of each representation and then judge which is greater. The challenge is fluency across forms, not new math.

Extracting features from each form

The same feature looks different depending on the representation:

Once each function's feature is a number, the comparison is just arithmetic.

Comparing intercepts and extrema

The same approach handles intercepts and maximums. For two parabolas, one given as $f(x) = -(x - 2)^2 + 9$ (vertex value $9$) and one graphed with a peak at height $6$, the first has the greater maximum. For $y$-intercepts, compare the constant term of an equation with the graph's $y$-axis crossing.

How the B.E.S.T. EOC examines this topic

• Multiple choice and multiselect. Decide which function has the greater rate, intercept, or maximum.
• Equation editor. Compute and report the difference between two functions' rates of change.
• Matching. Pair descriptions ("steeper," "higher start") with the correct function.

A clarifying idea: every representation encodes the same information, so the comparison is always possible once you translate. A table hides the slope until you compute a difference quotient; an equation hides the maximum until you find the vertex. The skill is knowing where each feature is stored in each form.

Why translating to one form makes the comparison reliable

Comparisons go wrong when students compare features that are not the same thing, for instance comparing a table's output at $x = 2$ with an equation's $y$-intercept. The fix is to decide which feature the question asks about, then read that one feature from both functions in the same units before comparing. If the question is about rate of change, get a slope from each; if it is about starting value, get a $y$-intercept from each. Because all four representations describe a single function, a feature computed from a table must match the same feature computed from that function's equation, so translating both to comparable numbers removes ambiguity. This discipline, name the feature, extract it from each form, then compare, is what the EOC rewards.

A note on growth: linear versus exponential

Some comparison items contrast a linear and an exponential function. Over a short interval the linear function may be ahead, but an exponential eventually overtakes any linear function because it multiplies rather than adds. If a table grows by a constant difference it is linear; if it grows by a constant ratio it is exponential. Recognizing the growth type from the table or equation lets you predict which function is larger far out, a common multiple-choice extension.

Try this

Q1. Function A: $f(x) = 2x + 10$. Function B: table $(0, 3), (1, 7), (2, 11)$. Which has the greater rate of change? [1 point]

• Cue. A's slope is $2$; B's is $4$, so B is greater.

Q2. Function A has $y$-intercept $5$ on a graph. Function B is $g(x) = 3x - 2$. Which has the greater $y$-intercept? [1 point]

• Cue. A ($5$) versus B ($-2$), so A.