Exercise 14 Page 244

We know that the amount of liquid fertilizer made depends on the amount of powdered fertilizer used, so different amounts of powdered fertilizer will never make the same amount of liquid fertilizer. Let's confirm this with a table of values.

Table

Following the recipe for making fertilizer, every time we mix $8$ teaspoons of powder with water, we make $5$ gallons of fertilizer.

Teaspoons of Powder, $p$	Gallons of Liquid, $\ell$
$0$	$0$
$8$	$5$
$16$	$10$
$24$	$15$

A function is a relationship that pairs each input value with exactly one output value. We can see from the table that the amount of fertilizer made is a function of the amount of powder used.

Equation

We know that the amount of liquid fertilizer $\ell$ made depends on the amount of powdered fertilizer $p.$ We need to find the constant of variation $k$ to define the relationship and write its equation. In this equation, $k$ represents how many teaspoons of powder it would take to create a certain amount of gallon of liquid fertilizer. We know that $5$ gallons of liquid fertilizer can be made from $8$ teaspoons of powdered fertilizer. Let's substitute these values into the equation above and solve for $k.$ Now that we know the constant of variation $k=\frac{5}{8},$ we can write a general equation to describe the relation between liquid and powder. With this equation, we know that $1$ teaspoon of powder makes $\frac{5}{8}$ gallon of liquid fertilizer.

Graph

Finally, let's graph the function by plotting the points from our table on a coordinate plane and connecting them with a line.