Compare properties of two functions represented in different ways (graph, table, equation, words), and build a function (linear or exponential) to model a relationship from a description or data.

What this topic is asking

The Functions category asks you to compare two functions given in different representations (graph, table, equation, words) and to build a function from a description or data (the F-IF, F-BF, and F-LE standards). On the Grade 10 MCAS this tests whether you can read the same feature, a slope, an intercept, a maximum, from whichever form it is presented in, and whether you can translate a situation into a linear or exponential model.

Comparing across representations

A function may be given as a graph, a table, an equation, or a verbal description, and the MCAS pairs two different forms and asks which has the larger slope, intercept, or maximum. The strategy is to extract the same feature from each:

• Y-intercept: the output at $x = 0$. In an equation $y = mx + b$ it is $b$; in a table it is the $y$ when $x = 0$; on a graph it is where the curve meets the y-axis.
• Rate of change (slope): in an equation it is the coefficient $m$; in a table it is the constant difference per unit step; on a graph it is the steepness.
• Maximum or minimum: read the vertex from an equation or graph, or the largest or smallest output in a table.

Once both functions are described by the same feature, the comparison is a direct numerical one.

Building a function from data

To model a relationship, first decide its type by checking how the outputs change:

• If equal steps in $x$ give a constant difference in $y$, the function is linear: $y = mx + b$, with $m$ the common difference per unit and $b$ the value at $x = 0$.
• If equal steps in $x$ give a constant ratio in $y$, the function is exponential: $y = A_0 b^x$, with $A_0$ the value at $x = 0$ and $b$ the common ratio.

For a table $x = 0, 1, 2$ with $y = 4, 12, 36$: the ratios are all 3, so $y = 4 \cdot 3^x$. For $y = 4, 7, 10$: the differences are all 3, so $y = 3x + 4$.

Average rate of change

For a non-linear function, the average rate of change over an interval $[a, b]$ is the slope of the secant line joining the endpoints:

For $f(x) = x^2$ from $x = 1$ to $x = 4$, it is $\dfrac{16 - 1}{3} = 5$. This is a recurring MCAS task, and the trap is forgetting to divide by the change in $x$: the rate is per unit input, not just the total change in output.

Average rate of change also lets you compare a curve with a line over the same interval. If a linear function has slope 4 and a quadratic has an average rate of change of 5 over $[1, 4]$, the quadratic is rising faster on that interval, even though its instantaneous steepness varies. A further comparison the MCAS likes: over a long enough interval, an exponential function's average rate of change exceeds that of any linear function, because the exponential eventually outgrows the line.

Try this

Q1. A line has slope 2; a table shows another line rising 3 per unit step. Which is steeper?

• Cue. The table's line, slope 3 versus 2.

Q2. Find the average rate of change of $f(x) = 2^x$ from $x = 0$ to $x = 3$.

• Cue. $\frac{8 - 1}{3} = \frac{7}{3}$.