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Amazing.
Word problems on the SAT can feel like little puzzles that make your brain do backflips, especially when test anxiety kicks in. But don’t worry! Once you know the tricks for spotting the difference between an exponential and a linear function—and whether they’re increasing or decreasing—you’ll breeze through these problems like a pro.
Let’s break down how to tell what kind of function you’re dealing with so you can ace those SAT Math questions without breaking a sweat.
Step 1: Spotting a Linear Function vs. an Exponential Function
First things first: what’s the difference between a linear function and an exponential function?
- A linear function changes at a constant rate. Think of it like adding or subtracting the same number every time. For example, if you earn $10 for every hour you work, that’s linear—it’s the same $10 added for each hour.
- An exponential function changes at a multiplicative rate. Instead of adding the same amount, you’re multiplying. It’s like when money grows with interest—your amount multiplies each time, and it gets bigger (or smaller) faster.
So, how do you tell which is which in a word problem? Let’s dive in!
Step 2: Keywords That Signal Linear Functions
If a word problem mentions a constant rate of change, it’s usually linear. Look out for keywords like:
- “Per year,” “per day,” or “per hour.”
- “Each time” or “for every.”
- Descriptions that involve “adding” or “subtracting” a set amount.
For example:
A tree grows 2 inches per year.
This is a linear situation because the tree’s height increases by the same amount—2 inches—each year. It’s consistent!
Now, is this an increasing linear function or a decreasing linear function? Easy! Since the tree’s height is going up, it’s increasing. If the problem described something shrinking or being taken away at a steady rate (like losing money every week), it would be decreasing.
Step 3: Spotting Keywords for Exponential Functions
Exponential functions can be a little trickier, but they have their own clues. You’ll know you’re dealing with an exponential function when you see phrases like:
- “Doubles,” “triples,” or “grows by a percentage.”
- “Multiplies” or “increases/decreases by a factor.”
- “Half-life” or anything that talks about things getting smaller by halves.
For example:
A population of bacteria doubles every hour.
This is an exponential function because the bacteria multiply (double) rather than just add a set number each hour.
To figure out if it’s an increasing exponential function or a decreasing exponential function, think about what’s happening: since the bacteria population is growing, it’s increasing. If you had something shrinking exponentially—like radioactive decay (where material halves over time)—that would be a decreasing exponential function.
Step 4: Practice with Examples
Let’s practice with some example problems to see how these tips work in action!
Example 1: The Falling Price
The price of a car decreases by $1,500 every year.
What do you think—linear or exponential?
- The key phrase here is “decreases by $1,500 every year.” That’s a consistent rate of change, which means it’s linear.
- Since the price is going down, it’s a decreasing linear function.
Example 2: The Super Savings Account
An investment grows by 5% each year.
Okay, now we’re talking about a percentage growth. Is that linear or exponential?
- Percentages usually mean you’re multiplying, which means it’s exponential.
- Since the investment is growing, it’s an increasing exponential function.
Example 3: The Shrinking Balloon
A balloon loses half its air every minute.
Half of something every time? That’s our clue that it’s exponential!
- Since the balloon is shrinking (losing air), it’s a decreasing exponential function.
Step 5: Test Anxiety Tip: Take a Breath and Look for Clues
When you’re feeling the pressure of SAT anxiety, it’s easy to overthink these problems. But remember, word problems are just stories, and you’re the detective. Here’s how to stay calm:
- Read the problem slowly: Take your time to spot keywords like “per,” “every,” or “percentage.”
- Ask yourself: is it adding or multiplying? If it’s adding or subtracting, it’s linear. If it’s multiplying or dealing with percentages, it’s exponential.
- Think about the direction: Is it going up (increasing) or down (decreasing)? That’ll tell you the type of function!
Step 6: Practice Makes Perfect
Now that you know the tricks for spotting these functions, practice is your best friend! The more problems you work through, the quicker you’ll be at identifying whether it’s linear or exponential—and whether it’s increasing or decreasing.
Final Thoughts
Word problems might seem tough at first, but once you know the clues, you can solve them like a pro. Remember to keep an eye out for keywords, take it one step at a time, and don’t let SAT anxiety mess with your focus. With these strategies, you’ll be able to tackle any word problem and determine whether it’s best modeled by a decreasing exponential function, increasing exponential function, decreasing linear function, or increasing linear function.
You’ve got this—just breathe, read carefully, and trust your skills. Good luck, and go crush that SAT!