Lesson 6: Functions as Models

New Concept

This lesson explores how functions serve as powerful models for real-world phenomena. We will learn to interpret the shape, slope, and concavity of graphs to understand the stories they tell and select appropriate basic functions to represent different scenarios.

What’s next

Soon, you'll work through interactive examples analyzing graph shapes. Then, you'll practice choosing the correct function models and solve problems involving absolute value.

Property

A graph that bends upward is called concave up, and one that bends down is concave down.

If a graph is concave up, its slopes are increasing. This describes a rate of change that is speeding up.

If a graph is concave down, its slopes are decreasing. This describes a rate of change that is slowing down.

Property

To model a situation, we can choose one of the eight basic functions based on the relationship between variables.

• The number of board-feet, $B$, is a function of the cube of the circumference, $c$: $B = k c^3$
• The optimal order size, $Q$, is a function of the square root of the annual demand, $D$: $Q = k \sqrt{D}$
• The intensity of light, $I$, is a function of the reciprocal of the square of your distance, $d$: $I = \frac{k}{d^2}$

To decide which basic function might model a set of data, we can plot the data.

Property

The distance between two points $x$ and $a$ is given by $|x - a|$.

Examples

• The statement "$x$ is five units from the origin" can be written using absolute value notation as $|x| = 5$. The solutions are $x=5$ and $x=-5$.
• The statement "$p$ is two units from 7" can be written as $|p - 7| = 2$. The solutions are $p=5$ and $p=9$.
• The statement "$a$ is within four units of $-3$" can be written as $|a - (-3)| < 4$, which simplifies to $|a + 3| < 4$. This means $a$ is between $-7$ and $1$.

Explanation

Absolute value is a tool for measuring distance on a number line without worrying about direction. The expression $|x - a|$ asks the question, "How far apart are $x$ and $a$?" The result is always a positive number.

Property

The equation $|ax + b| = c$ (where $c > 0$) is equivalent to:

Examples

• To solve $|x - 3| = 8$, we set up two equations: $x - 3 = 8$ or $x - 3 = -8$. The solutions are $x = 11$ and $x = -5$.
• To solve $|2y + 5| = 11$, we set up two equations: $2y + 5 = 11$ or $2y + 5 = -11$. The solutions are $y = 3$ and $y = -8$.
• To solve $|\frac{z}{3} - 1| = 4$, we set up two equations: $\frac{z}{3} - 1 = 4$ or $\frac{z}{3} - 1 = -4$. The solutions are $z = 15$ and $z = -9$.

Explanation

To solve an absolute value equation, you split it into two separate linear equations. This is because the expression inside the absolute value bars could be either positive or negative, and both would result in the same positive value.

Property

Suppose the solutions of the equation $|ax + b| = c$ are $r$ and $s$, with $r < s$. Then:

1. The solutions of $|ax + b| < c$ are $r < x < s$.
2. The solutions of $|ax + b| > c$ are $x < r$ or $x > s$.

Examples

• To solve $|x - 5| < 3$, we find solutions between the points where $|x-5|=3$, which are $x=2$ and $x=8$. The solution is $2 < x < 8$.
• To solve $|2y + 1| \geq 7$, we find solutions outside the points where $|2y+1|=7$, which are $y=3$ and $y=-4$. The solution is $y \leq -4$ or $y \geq 3$.
• The solutions to $|z - 10| \leq 5$ are all numbers whose distance from 10 is 5 or less. This corresponds to the interval $5 \leq z \leq 15$.

Explanation

For absolute value inequalities, "less than" means the distance is small, so the solutions are between two points. "Greater than" means the distance is large, so the solutions are outside those two points, heading in opposite directions.