The number of species of coastal dune plants in Australia 2...

The number of species of coastal dune plants in Australia $2 E$ decreases as the latitude, in $^{\circ} \mathrm{S}$, increases. There are 34 species at $11^{\circ} \mathrm{S}$ and 26 species at $44^{\circ} \mathrm{S} .^{19}$ $3 i$ (a) Find a formula for the number, $N$, of species of coastal dune plants in Australia as a linear function of the latitude, $l$, in $^{\circ}$ S. (b) Give units for and interpret the slope and the vertical intercept of this function. (c) Graph this function between $l=11^{\circ} \mathrm{S}$ and $l=$ $44^{\circ} \mathrm{S} .$ (Australia lies entirely within these latitudes.)

The number of species of coastal dune plants in Australia $2 E$ decreases as the latitude, in $^{\circ} \mathrm{S}$, increases. There are 34 species at $11^{\circ} \mathrm{S}$ and 26 species at $44^{\circ} \mathrm{S} .^{19}$
$3 i$
(a) Find a formula for the number, $N$, of species of coastal dune plants in Australia as a linear function of the latitude, $l$, in $^{\circ}$ S.
(b) Give units for and interpret the slope and the vertical intercept of this function.
(c) Graph this function between $l=11^{\circ} \mathrm{S}$ and $l=$ $44^{\circ} \mathrm{S} .$ (Australia lies entirely within these latitudes.)

Key Concepts

Linear Functions

A linear function is a mathematical representation of a relationship between two variables that changes at a constant rate. It can be expressed in the form y = mx + b, where m represents the constant rate of change (slope) and b represents the value when the independent variable is zero (vertical intercept). This type of model is fundamental in understanding and predicting how one variable changes in response to another in a consistent manner.

Slope

The slope of a linear function indicates the rate at which the dependent variable changes with respect to the independent variable. It shows the change in the output for every one unit increase in the input. In real-world scenarios, its units are determined by the units of the dependent variable divided by the units of the independent variable, thus providing meaningful interpretation in context.

Vertical Intercept

The vertical intercept of a linear function is the value of the function when the independent variable is zero. It provides a starting point or baseline for the relationship before any changes in the independent variable occur. In practical contexts, it represents an extrapolated value that may be used for understanding initial conditions or reference points.

Graphing Linear Equations

Graphing a linear function involves plotting points that satisfy the equation and drawing a straight line through these points. This visual representation helps to illustrate the relationship between the variables, including the slope and vertical intercept, and is useful for understanding and analyzing the behavior of the function over a specified range.

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