For the following exercises, identify whether the statement...

For the following exercises, identify whether the statement represents an exponential function. Explain.
For each training session, a personal trainer charges his clients $\$ 5$ less than the previous training session.

Key Concepts

Exponential Function

An exponential function is characterized by a constant multiplicative rate of change, meaning each value is obtained by multiplying the previous value by a fixed factor. This type of function typically represents growth or decay processes where changes occur by percentages rather than fixed amounts.

Arithmetic Sequence

An arithmetic sequence (or arithmetic function when expressed in a function format) involves adding or subtracting a fixed amount between consecutive terms. In scenarios where a quantity is increased or decreased by a constant value, the change is linear rather than exponential.

Transcript

00:01 Okay, so for each training session, a personal trainer charges his client's $5 or less than the previous training session.

00:10 And we have to figure out if it's linear or if it's exponential or not.

00:17 So let's think of what this means exactly.

00:19 So if this guy is charging $5 or less every single time consistently, right, let's say let's start off with maybe it's $100 first session, right? this guy was charging someone $100 for a trading session.

00:35 Next time, it would have been $95, right? because it's $5 less.

00:39 And then it would have been $90 than $85 and so on and so on, you know, all the way until, you know, however long.

00:48 But notice one thing.

00:50 Every single time it's the same rate of change.

00:52 Every single time it's five, right? minus five, minus five, minus five, minus five, minus five, so on, so on, you know, for however long it went.

01:01 Now, why is it important? well, the reason why is because we can actually graph something like this and see what type of line it makes.

01:07 Is it a curve line? is it a straight line? like, what is it exactly? and from there, we can figure out, okay, is this exponential growth, exponential decay? is it linear growth, linear decay? like, what's going on over here? so let's try to figure this out.

01:20 So remember, what are our two graphs that we're usually dealing with in this section? it's either linear or exponential.

01:27 So linear lines look something like this.

01:30 And exponential lines look something like this.

01:33 And note that for both of these, i'm drawing them downwards because the problem kind of is relating to decay in the first place, right? it's lower numbers every single time.

01:45 So what's so special about linear lines? well, linear, you can see that for any given point, it's a constant rate of change...

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