Interpret slope and intercept in context

Learning goals

Identifies slope and intercept from a graph or equation.
Interprets slope as a rate of change with units.
Interprets the intercept as an initial or baseline value.
Assesses whether the interpretation is meaningful in context.

Difficulty: 1 – 5

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Interpret slope and intercept in context 1

A bike-rental shop models the total cost by $C(h)=8h+15$, where $h$ is the number of rental hours and $C(h)$ is the cost in dollars. Interpret the slope and the y-intercept in this situation. Include units.

Hints

- Connect the coefficient of $h$ with how the cost changes when rental time increases. - Evaluate what the model says when the number of rental hours is $0$. - Match each numerical parameter with the units it has in context.

Solution

1. The slope is $8$. This means the cost increases by $\$8$ for each additional hour of rental time. 2. The y-intercept is $15$. Since $C(0)=15$, it represents a fixed $\$15$ charge before any rental hours are added.

Answer

The slope means $\$8$ per rental hour. The y-intercept means there is a fixed $\$15$ charge at $0$ hours.

Interpret slope and intercept in context 2

A school fundraiser has already collected $\$120$ in donations. Tickets cost $\$15$ each. The total money collected is $F(t)=120+15t$ dollars, where $t$ is the number of tickets sold. Explain what the slope $15$ means. Give its units.

Hints

- What does the input count? - What does an increase of one input unit do to the output?

Solution

1. Increasing $t$ by one adds $15$ to $F(t)$. 2. The slope means the fundraiser collects $\$15$ for each additional ticket, measured in dollars per ticket.

Answer

Each additional ticket brings in $\$15$; the rate is $15$ dollars per ticket.

Interpret slope and intercept in context 3

A bathtub has $18$ gallons of water and drains steadily for the first $4$ minutes. The amount left after $t$ minutes is $W(t)=18-3t$. What does $18$ mean? What does $-3$ mean, including units?

Hints

- Which number gives the amount before any time passes? - What happens to the output when time increases by one minute? - What units should a change per minute have?

Solution

1. At $t=0$, the tub has $W(0)=18$ gallons of water, so $18$ is the starting amount. 2. Each minute the water decreases by $3$ gallons. The slope is $-3$ gallons per minute.

Answer

$18$ is the starting amount in gallons. $-3$ means the amount falls by $3$ gallons per minute.

Interpret slope and intercept in context 4

A market analyst models the supply price for a product by $p(x)=1.5x+20$, where $x$ is the number of units and $p(x)$ is the price per unit in dollars. A $\$5$ subsidy per unit is paid directly to producers, so producers can request $\$5$ less from buyers for the same total payment. a) Write the new supply-price function $p_{\mathrm{sub}}(x)$. b) Explain why the subsidy is represented by a downward vertical shift. c) Find and interpret the new y-intercept within this model.

Hints

- Determine how much buyers must pay after the producer receives the subsidy. - Subtracting the same constant from every output creates a vertical shift. - Evaluate the new function at $x=0$. - Interpret the intercept only within the stated model.

Solution

1. Subtract the subsidy from every modeled price: $p_{\mathrm{sub}}(x)=p(x)-5=1.5x+15$. 2. Because every output decreases by the same amount, every point on the graph moves down $5$ units. 3. At $x=0$, $p_{\mathrm{sub}}(0)=15$, so the y-intercept is $(0, 15)$. 4. Within the algebraic model, the intercept means the modeled price is $\$15$ when the quantity is $0$. It should not automatically be interpreted as a real-world minimum operating price without additional economic assumptions.

Answer

a) $p_{\mathrm{sub}}(x)=1.5x+15$ b) The subsidy subtracts $5$ from every output, so the graph shifts down $5$ units. c) The y-intercept is $(0, 15)$, representing a modeled price of $\$15$ at a quantity of $0$.

Interpret slope and intercept in context 5

A school club models its profit from selling event tickets by $P(n)=12n-180$, where $n$ is the number of tickets sold and $P(n)$ is the profit in dollars. a) Interpret the slope. b) Interpret the y-intercept, including why a negative value is meaningful here. c) State a reasonable contextual domain for $n$.

Hints

- Ask what one additional ticket changes in the model. - Evaluate the model conceptually at zero tickets and interpret the sign of the result. - Use the fact that $n$ counts physical tickets when deciding which input values make sense.

Solution

1. The slope is $12$, so each additional ticket sold increases profit by $\$12$. 2. The y-intercept is $-180$. At $n=0$, the club has a profit of $-\$180$, meaning it has $\$180$ in upfront costs before any tickets are sold. 3. Ticket counts cannot be negative or fractional, so a reasonable domain is the nonnegative integers.

Answer

a) Profit increases by $\$12$ for each ticket sold. b) The y-intercept $-\$180$ represents $\$180$ in upfront costs before any tickets are sold. c) $n$ should be a nonnegative integer.

Interpret slope and intercept in context 6

The graph shows a plant's height in the weeks after it was repotted. How tall was the plant when it was repotted, and how many centimeters does the model add per week? Show your rate calculation using the axis values.

Hints

- Read the height when the number of weeks is zero. - How much does the graph rise over the marked time interval?

Solution

1. The graph meets the height axis at $(0,12)$, so the plant's starting height was $12\,\text{cm}$. 2. It also passes through $(2,18)$. The rate is $\frac{18-12}{2-0}=3$ centimeters per week.

Answer

Height when repotted: $12\,\text{cm}$. The rate is $\frac{18-12}{2-0}=3\,\text{cm}$ per week.

Interpret slope and intercept in context 7

A kayak rental costs $C(h)=12+4h$ dollars for $h$ hours. The rental shop accepts bookings of at least $2$ hours. What is the y-intercept of the equation? Does it describe an available rental? Explain.

Hints

- The y-intercept uses an input of zero. - Compare that input with the shop's minimum rental time.

Solution

1. At $h=0$, the equation gives $C(0)=12$, so the y-intercept is $(0,12)$. 2. The shop rents kayaks only for $h\ge2$, so $h=0$ is not an available booking. The intercept belongs to the algebraic line but not the meaningful graph of the rental model.

Answer

The y-intercept is $(0,12)$. No; a zero-hour rental is not offered.

Interpret slope and intercept in context 8

A basketball player keeps track of their total points scored in practice games. From game $5$ through game $10$, a line fits the running totals: $P(g)=5g-20$, where $g$ is the number of games played. What total does the line predict at $g=0$? Why should that prediction not be treated as real?

Hints

- Substitute zero into the fitted rule. - Can the stated quantity actually take that value? - Were any games near zero used to create the model?

Solution

1. The equation predicts $P(0)=5\cdot0-20=-20$ points. 2. A running total of $-20$ points is impossible, and the line was fitted only to games $5$ through $10$. Its y-intercept is a prediction outside the data, not a real total.

Answer

The predicted y-intercept is $-20$ points. A running total cannot be negative, and the line was fitted only to games $5$ through $10$.

Interpret slope and intercept in context 9

For its first four hours in the sun, a pile of snow is modeled by $H(t)=30-2t$, where $H$ is height in centimeters and $t$ is hours. The formula gives $H(20)=-10$. Should you use this prediction for hour $20$? Explain.

Hints

- What time interval was the model based on? - Is the predicted physical quantity possible?

Solution

1. The value $-10$ is a result of extending the straight-line rule far beyond the first four hours. 2. A snow pile cannot have a negative height. The model is not reliable for this prediction; the pile would reach zero before that if the rate persisted.

Answer

No. A negative snow-pile height is impossible, and hour $20$ is far outside the four-hour period modeled.

Interpret slope and intercept in context 10

Two streaming services charge a monthly fee plus a price for each movie you rent. For $n$ movies in one month, the costs are: StreamClub: $S(n)=10+6n$ dollars FilmPass: $F(n)=24+3n$ dollars Which service has the lower monthly fee? Which charges less for each movie? Which costs less for $2$ movies and for $8$ movies?

Hints

- What does each service charge before you rent any movies? - What does the cost go up by when you rent one more movie? - Compare the total costs at each requested movie count.

Solution

1. StreamClub's monthly fee is $\$10$, below FilmPass's $\$24$. FilmPass adds $\$3$ per movie, below StreamClub's $\$6$. 2. For $n=2$, $S(2)=22$ and $F(2)=30$, so StreamClub costs less. 3. For $n=8$, $S(8)=58$ and $F(8)=48$, so FilmPass costs less.

Answer

Lower monthly fee: StreamClub. Lower cost per movie: FilmPass. For $2$ movies, StreamClub costs $\$22$, less than $\$30$. For $8$ movies, FilmPass costs $\$48$, less than $\$58$.

Interpret slope and intercept in context 11

Two monthly phone plans are modeled by linear functions, where $x$ is data usage in gigabytes and $K(x)$ is the monthly cost in dollars. Plan A: $K_A(x)=2x+20$ Plan B: $K_B(x)=3x+5$ 1. Identify $m$ and $b$ for each plan and explain what each parameter means in context. 2. Determine when Plan A costs less than Plan B.

Hints

- In $K(x)=mx+b$, identify the part that changes with usage and the fixed part. - Use the units to interpret the slope and intercept. - Write a strict inequality to represent “Plan A costs less.”

Solution

1. For Plan A, $m=2$ and $b=20$. The rate is $\$2$ per gigabyte, and the monthly base charge is $\$20$. 2. For Plan B, $m=3$ and $b=5$. The rate is $\$3$ per gigabyte, and the monthly base charge is $\$5$. 3. Compare the plans: $2x+20<3x+5$. Subtracting $2x$ and $5$ gives $15<x$. Therefore, Plan A costs less when usage exceeds $15$ GB.

Answer

1. Plan A: $m=2$ dollars per GB, $b=\$20$. Plan B: $m=3$ dollars per GB, $b=\$5$. 2. Plan A costs less when $x>15$ GB.

Interpret slope and intercept in context 12

The graph shows the height $h$, in centimeters, of a candle $t$ hours after it is lit. The model is linear over the time shown. a) What does the y-intercept mean in this context? b) Find and interpret the slope, including units. c) According to the model, what is the candle's height after $5$ hours?

Hints

- Read the graph where $t=0$ to interpret the intercept. - Use two clear graph points and compare vertical change with horizontal change. - The sign of the slope should agree with whether the candle gets taller or shorter over time.

Solution

1. From the graph, the y-intercept is $12$, so the candle is $12\,\text{cm}$ tall when it is lit. 2. Using the points $(0,12)$ and $(2,9)$, the slope is $\frac{9-12}{2-0}=-\frac{3}{2}=-1.5$. The candle's height decreases by $1.5\,\text{cm}$ per hour. 3. Continuing the linear model to $t=5$ gives $h=12-1.5\cdot5=4.5$. The predicted height is $4.5\,\text{cm}$.

Answer

a) The candle's initial height is $12\,\text{cm}$. b) The slope is $-1.5\,\text{cm per hour}$, meaning the candle loses $1.5\,\text{cm}$ of height each hour. c) $4.5\,\text{cm}$

Interpret slope and intercept in context 13

A gym charges a one-time signup fee of $\$35$ plus the same membership charge each month. After $6$ months, a member has paid a total of $\$185$. a) Write a linear model $C(m)$ for the total amount paid after $m$ months. b) Interpret both parameters of your model in context. c) Another member says the same plan should cost $\$285$ after $10$ months. Is that consistent with your model? Justify your answer.

Hints

- Identify which given amount belongs to month $0$. - The total increase after several months comes from repeated equal monthly charges. - After writing the model, test the final claim by evaluating it at the stated month.

Solution

1. The y-intercept is the signup fee, so $C(0)=35$. 2. The monthly charge is $\frac{185-35}{6}=25$ dollars per month. 3. The model is $C(m)=25m+35$. The slope $25$ is the monthly membership charge, and the y-intercept $35$ is the one-time signup fee. 4. $C(10)=25\cdot10+35=285$, so the second member's statement is consistent with the model.

Answer

a) $C(m)=25m+35$ b) The slope is a $\$25$-per-month charge, and the y-intercept is the $\$35$ signup fee. c) Yes. The model gives $C(10)=\$285$.

Interpret slope and intercept in context 14

An energy company models a family of annual plans by $K_p(x)=px+(300-2000p)$, where $x$ is annual usage in kWh, $p$ is the price per kWh in dollars, and $K_p(x)$ is annual cost in dollars. 1. For a plan with $p=0.12$, identify the slope and y-intercept and interpret both in context. 2. Show that all graphs in the family pass through one common point. Find the point and interpret it in context. 3. A customer uses $3500\,\text{kWh}$ per year. Determine whether a higher or lower value of $p$ gives this customer a lower annual cost.

Hints

- Substitute the specified value of $p$ before interpreting the slope and intercept. - Rewrite the family so the coefficient of $p$ is easy to see. - Find the usage value that makes the $p$-dependent part equal to $0$. - For the final comparison, substitute $3500$ and see how the cost changes as $p$ changes.

Solution

1. With $p=0.12$, $K_{0.12}(x)=0.12x+60$. The slope $0.12$ means each additional kWh adds $\$0.12$ to the modeled annual cost. The y-intercept $60$ means the model gives an annual cost of $\$60$ at zero usage. 2. Rewrite the family as $K_p(x)=p(x-2000)+300$. At $x=2000$, the term containing $p$ is $0$, so every plan gives $K_p(2000)=300$. The common point is $(2000,300)$: every plan costs $\$300$ at $2000\,\text{kWh}$. 3. At $x=3500$, $K_p(3500)=300+1500p$. This increases as $p$ increases, so a lower value of $p$ gives the lower annual cost.

Answer

1. For $p=0.12$, slope $=0.12$ dollars per kWh and y-intercept $=60$ dollars. The slope is the added cost per additional kWh; the intercept is the modeled cost at zero usage. 2. Common point: $(2000,300)$. At $2000\,\text{kWh}$, every plan costs $\$300$. 3. A lower value of $p$ gives the lower annual cost at $3500\,\text{kWh}$.

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Interpret slope and intercept in context

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