​Parts of an Expression/Equation Review

​Parts of an Expression/Equation Review

By Melissa Anders

↗️ If that number is positive,
the quantity is INCREASING.

↘️ If that number is negative,
the quantity is DECREASING.

🗻 Slope

📍 Constant

The constant term tells you the starting value.

Read the questions CLOSELY!

• The slope is referred to as the "rate of change."

• The y-intercept and the constant are the same thing

  • They are referred to as the "starting value" or the "initial amount."

• If the answer choice talks about "per hour" or "per minute" or "per day," it's probably asking you about slope.

Units Matter!

Use them like clues 🕵️

🗣️ Say It Out Loud!

Is the situation going up or down over time?

If slope < 0, it's going down ↘️
If slope > 0, it's going up ↗️

🤔Ask yourself this question every time...

⭐ Two More Tips! ⭐

Multiple Choice

A botanist models the height of a fast-growing plant using the function
$$H(t)=2t+15$$ ,
where t is the number of days after the plant was planted and H(t) is the height in centimeters.

Which statement is true based on the model?

• The plant grows 2 cm per day

• The plant grows 15 cm per day

• The plant shrinks 2 cm per day

• The plant shrinks 15 cm per day

Multiple Choice

A roofer models the temperature of roof shingles using the function
$$S(t)=-5t+140$$ ,
where t is the number of hours after sunrise and S(t) is the surface temperature in °F.

Which statement is true based on the model?

• The temperature decreases by 5°F per hour

• The temperature increases by 5°F per hour

• The temperature decreases by 140°F per hour

• The temperature increases by 140°F per hour

Multiple Choice

A mechanic models the temperature of an engine after being turned off using the function
$$E(t)=-4t+190$$ ,
where t is the number of minutes after the engine is turned off and E(t) is the temperature in °F.

Which statement is true based on the model?

• The engine started at 190°F when it was turned off.

• The engine loses 190°F every minute.

• The engine started at 4°F when it was turned off.

• The engine loses 4°F every minute.

Multiple Choice

A delivery driver tracks the number of packages left on the truck using the function
$$P(t)=-7t+84$$ ,
where t is the number of hours after leaving the warehouse and P(t) is the number of packages remaining.

Which statement is true based on the model?

• The driver started with 84 packages.

• The driver delivers 84 packages each hour.

• The driver started with 7 packages.

• The driver delivers 7 packages per minute.

The y-intercept=the starting amount

It’s the value when x = 0 (zero pounds, zero hours, zero miles, zero treats).

👉 Ask Yourself... “What would the cost or starting amount be before adding any weight/time/miles?”

The y-intercept is NOT the rate!

🤷‍♀️ Don't confuse the two.

• Slope: cost per pound/hour/mile

• Intercept: cost when x = 0

If it’s attached to the x, it’s the rate.

If it’s by itself, it’s the starting cost.

Word Problems Key Words

In real-world cost problems, the y-intercept is almost always a fee.

• 👀 Look for words like...

  • “initial fee”

  • “starting cost”

  • “flat rate”

  • “base charge”

• These almost always describe the y-intercept.

Example:

If a restaurant charges you $6.50 for extra cookies 🍪, and each cookie adds money, then the restaurant must have charged something before the cookies were added.

🧠 Use the table to think backwards to zer0

That “something” is the y-intercept.

Wrong answers are usually "distractors"

• Common traps:

  • Picking the rate instead of the intercept

  • Choosing a value from the table

  • Picking a quantity (“number of treats”) instead of a cost

"Distractors" are answer choices put there because they are common mistakes students make...don't let the test trick you.

Multiple Choice

A pet supply store charges customers based on the number of dog treats they purchase.
The table shows the total cost for different amounts of treats:

The store models these costs with a linear function.

What does the y-intercept represent in this context?

• The cost per treat

• The total cost for 10 treats

• The initial fee charged before adding the cost per treat

• The number of treats purchased

Multiple Choice

A landscaping company charges customers based on the number of hours spent mowing their lawn.
The table shows the total cost for different job lengths.

The company models these costs with a linear function.

What does the y-intercept represent in this context?

• The hourly mowing rate

• The initial service fee charged before hourly costs

• The total cost for a 3.5-hour job

• The number of hours worked

Multiple Choice

A taxi service charges riders based on the number of miles traveled.
The table shows the cost of several trips.The company models these costs with a linear function.

What does the y-intercept represent in this context?

• The taxi’s cost per mile

• The number of miles in a trip

• The base fare charged before any miles are driven

• The total cost of a 6-mile ride

y-intercept = the amount at ⏱️ time zero

Whatever the “start of the story” is = the intercept.

• Time zero could mean:

  • at birth

  • at adoption

  • when planted

  • when purchased

  • when the clock starts

The y-intercept is always the “starting amount”

If time = 0 → look at the constant.

Money at the beginning Miles already traveled

Water already in the tank Temperature at time zero

Pages already read Initial concentration

These can’t be y-intercepts, because they aren't STARTING AMOUNTS

For example:

“Maximum weight”

“Total time”

“Distance traveled later”

Eliminate answers that describe impossible meanings

Multiple Choice

A student models the amount of money in her savings account with
$$S(t)=25t+120$$ ,
where t is the number of weeks she has been saving.

What does the y-intercept represent?

• The amount she deposits each week

• The total amount she saves after 120 weeks

• The amount of money she had saved before she began adding weekly deposits

• The number of weeks it takes to reach $120

Multiple Choice

A scientist uses the model
$$T(t)=-4t+92$$ ,
where t is the number of minutes after removing a metal rod from a furnace.

What does the y-intercept represent?

• The rod cools at a rate of 4°F per minute

• The number of minutes until the rod reaches 92°F

• The rod’s temperature when it was removed from the furnace

• The rod’s lowest possible temperature

Multiple Choice

A cell phone company models a customer’s monthly bill using
$$C(t)=0.12t+40$$ ,
where t is the number of texts sent.

What does the y-intercept represent?

• The cost per text message

• The cost of sending 40 texts

• The base monthly fee before paying for any texts

• The total cost after sending 40 texts

The slope always means “how much it changes per one unit.”

• Look for:

  • per hour

  • per mile

  • per customer

  • per day

  • per movie

If the answer uses per, it’s probably slope.

The slope is attached to the variable.

Eliminate answer choices about fees or starting amounts.

If the answer mentions:

• “initial fee”

• “starting cost”

• “flat charge”

  …it’s not the slope.

Those are y-intercepts, not slope.

Multiple Choice

A gym charges a sign-up fee and a cost per month.
The function:
M(x) = 12x + 35
can be used to determine the total cost for x months of membership.

What is the meaning of the slope?

• The gym charges $12 as an initial fee.

• The gym charges $35 as an initial fee.

• The cost increases by $12 per month.

• The cost increases by $35 per month.

Multiple Choice

A parking garage charges a flat entry fee plus an hourly rate.
The function:
P(x) = 4.5x + 8
represents the total cost for x hours of parking.

What is the meaning of the slope?

• The garage charges $4.50 for each hour parked.

• The garage charges $8 for each hour parked.

• The garage charges a $4.50 entry fee.

• The garage charges an $8 entry fee.

Multiple Choice

A streaming service charges a monthly subscription fee and a cost per movie rented.
The function:
C(x) = 3x + 14
shows the total cost for renting x movies in one month.

What is the meaning of the slope?

• The cost increases by $14 for each movie rented.

• The cost increases by $3 for each movie rented.

• The service charges a $14 monthly fee.

• The service charges a $3 monthly fee.