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# Function Notation and Linear Functions

## If y=mx+b and y=f(x) then f(x)=mx+b

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Practice Function Notation and Linear Functions

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Growing Kids
Teacher Contributed

## Real World Applications – Algebra I

### Topic

How can we represent a kid’s growing height as a linear relationship?

### Student Exploration

Most doctors agree that the “normal” growth rate for children after the age of 2 is about \begin{align*}2 \ \frac{1}{2} \ inches\end{align*} or 6 centimeters per year until adolescence. Let’s represent this as a linear relationship using inches.

Let’s say a kid at 2 years old is 3 feet tall, or 36 inches. Using the information given, this kid will be 38.5 inches tall when 3 years old. Let’s write an equation representing this relationship using these two data points.

Given the information about the heights, we’d first have to calculate the slope (even though that was given to us). The slope would be,

\begin{align*}m &= \frac{(\text{the difference in height})}{(\text{the difference in age})}\\ m &= \frac{(38.5 - 36)}{(3 - 2)} = 2.5 \ inches \ per \ year\end{align*}

Now let’s use one of our data points and the slope to find the equation to represent this relationship. We’re going to use the slope-intercept form to substitute what we know so far.

\begin{align*}y &= mx + b\\ 36 &= (2.5)(2) + b \ \text{Now let’s solve for} \ “b.”\\ 36 &= 5 + b\\ 31 &= b\end{align*}

Our equation is: \begin{align*}y = 2.5x + 31\end{align*}

Now, this equation represents the linear relationship of a growing child after the age of 2. Looking at the equation, 31 is \begin{align*}b\end{align*}. This means that the \begin{align*}y-\end{align*}intercept is 31 inches. This can’t make sense, because then this would mean that a child was born at 31 inches! This also wouldn’t make sense when a kid hits puberty, because his/her growth spurt would be a lot faster!

Now let’s look at this linear relationship as a function. As you read from the concept, the \begin{align*}f(x)\end{align*} is the output. We can rewrite this relationship as \begin{align*}f(x) = 2.5x + 31\end{align*}. We can use this function to determine height at different ages.

If we were to find \begin{align*}f(5)\end{align*}, this means that we need to find the height of the child at 5 years old. Let’s figure it out:

\begin{align*}f(5) &= 2.5(5) + 31\\ f(5) &= 7.5 + 31\\ f(5) & = 38.5\end{align*}

This means that at 5 years old, the child will be 38.5 inches tall.

What’s \begin{align*}f(7)\end{align*} and what does it mean?

### Extension Investigation

Try asking a family member how tall you were at two different ages in your life, and practice finding the rate of change, or the slope between these two points. Would this equation make sense? Why or why not? Would this equation apply when you’re over 30 years old? Would you be getting taller at that age?

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