Compare properties of linear, quadratic, and exponential functions represented in different ways, and identify the family that models a situation (TN A1.F.IF.D.9, A1.F.LE.A.3).

What this topic is asking

Standards A1.F.IF.D.9 and A1.F.LE.A.3 ask you to compare the three function families of Algebra I, linear, quadratic, and exponential, across tables, graphs, and equations, and to know that an exponential eventually outgrows a linear (or even a quadratic). The graded skills are identifying the family from a representation and comparing growth.

Identifying the family from a table

The pattern in the outputs names the family.

Comparing growth rates

Standard A1.F.LE.A.3 makes one comparison explicit: a quantity growing exponentially eventually exceeds one growing linearly or quadratically, regardless of the starting values or rates. A line or parabola may lead early, but the exponential's multiplying steps compound and overtake it. The test shows this with tables (the exponential column passes the others) or two-plan word problems.

How TNReady examines this topic

• Multiple choice. Identify the family from a table, graph, or context, or choose which function grows faster eventually.
• Multiple select. Choose all correct comparisons of two functions in different forms.
• Inline choice. State whether a relationship is linear, quadratic, or exponential.

A clarifying idea is that this standard ties the module together: it reuses the linear-versus-exponential test from exponential functions, the parabola shape from graphing quadratics, and the constant-slope idea from writing linear functions.

Why each family has a signature

The reason the families look different in a table comes from how each builds its next output. A linear function adds the slope each step, so the gaps between outputs stay equal, that is the constant first difference. A quadratic function's rate of change itself changes at a steady pace, so the gaps grow by a constant amount, which is why the second differences are constant. An exponential function multiplies by a fixed factor, so each output is a constant proportion of the previous one, giving constant ratios. Knowing the mechanism, not just memorizing the table tests, lets you classify even a partial or messy table: compute differences first, and if those are not constant, compute ratios. Comparing functions across different representations (a graph versus an equation versus a table) is then a matter of extracting the same signature from each, for instance reading the constant ratio off a table and matching it to the base $b$ in an equation $y = ab^x$.

Try this

Q1. Outputs $5, 10, 20, 40$ for inputs $0,1,2,3$: which family? [1 point]

• Cue. Ratios all $2$, so exponential ($y = 5(2)^x$).

Q2. Which eventually grows larger: $y = 1000 + 200x$ or $y = 10(2)^x$? [1 point]

• Cue. The exponential $y = 10(2)^x$ eventually exceeds the linear, despite starting much smaller.