3.6 Functions from their graphs

A graph turns many numbers into a shape, but the shape is useful only when you read its axes and units. A rising line might mean increasing distance, cost, or error; those interpretations lead to different conclusions. Graph interpretation is needed whenever a question gives a plot instead of a formula or asks you to identify a rule from its behaviour.

This is intermediate work that combines earlier function ideas. OpenStax's lesson on graph behaviour covers slopes and turning points. Its function-notation lesson connects graphs with tables and formulas.

A basic idea: axes before shape

On a graph of distance from home against time, a flat segment means the distance is unchanged. It does not tell you whether the traveller is resting or moving in a circle around home. A downward segment means the traveller is getting closer to home; it does not mean time is running backwards. Read the horizontal variable, vertical variable, units, and scale before interpreting any line.

A quick tour of graph clues

For two graph points $(x_1,y_1)$ and $(x_2,y_2)$ with $x_1\ne x_2$, their average rate of change is $(y_2-y_1)/(x_2-x_1)$. A constant rate suggests a line. Constant multiplicative change across equal input steps suggests an exponential. A symmetric U-shape, upright or upside down, suggests a quadratic, while repetition suggests a periodic rule.

These are clues, not proofs that a graph has one unique formula. A few plotted points can fit many rules. If a question offers candidate functions, test their values and features against the evidence shown.

Application 1: read a journey

The figure shows distance from home during a four-hour journey. How long is the traveller $4$ km from home, and what are the outward and return rates of change in distance?

1. Read the flat section. Distance is $4$ km from hour $2$ to hour $3$, a duration of $1$ hour.
2. Find the outward slope. Distance rises from $0$ to $4$ km over $2$ hours, so its rate of change is $(4-0)/(2-0)=2$ km per hour.
3. Find the return slope. Distance falls from $4$ to $0$ km between hours $3$ and $4$, so the slope is $(0-4)/(4-3)=-4$ km per hour.

The negative slope says distance from home is shrinking. It is not a negative travel speed.

Application 2: identify a rule from values

The table gives four input-output pairs. Choose among a linear, quadratic, and exponential rule, then predict the next value under that choice.

1. Compare differences. The successive differences are $2$, $6$, and $18$, so a linear rule with constant difference does not fit. The second differences, $4$ and $12$, are not constant either, so no quadratic fits.
2. Compare ratios. Each output is $3$ times the previous one. That points to $f(x)=3^x$, which gives all four values.
3. Predict within the model. $f(4)=3^4=81$. The ratios suggest an exponential among the offered families; the four values alone cannot guarantee that every future value follows it.

The last distinction matters: identifying a model from limited evidence is different from proving a rule for all inputs.