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Interpret slope and intercept contextually

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.

Click problems to add them to your worksheet.

Interpret slope and intercept contextually 1
The amount of water in a container is modeled by \(V(t)=40-3t\), where \(t\) is the number of minutes after draining begins and \(V(t)\) is the amount of water in gallons. What do the initial value and the rate of change mean in this situation? Include units.

Hints

- Think about what the model gives when the elapsed time is zero. - The coefficient multiplying time tells how the output changes for each additional minute. - Interpret the sign of the rate in terms of whether the amount is increasing or decreasing.

Solution

In \(V(t)=40-3t\), the constant term \(40\) is the value when \(t=0\), so the container starts with \(40\,\text{gal}\) of water. The coefficient of \(t\) is \(-3\), so the amount changes by \(-3\,\text{gal}\) each minute. In context, the container loses \(3\,\text{gal}\) of water per minute.

Answer

Initial value: \(40\,\text{gal}\), the amount of water when draining begins. Rate of change: \(-3\,\text{gal/min}\), meaning the amount decreases by \(3\,\text{gal}\) each minute.
Interpret slope and intercept contextually 2
A parking garage charges according to the table. <table><tr><th>Hours \(h\)</th><th>Cost \(C\)</th></tr><tr><td>\(0\)</td><td>\(\$6\)</td></tr><tr><td>\(1\)</td><td>\(\$8\)</td></tr><tr><td>\(2\)</td><td>\(\$10\)</td></tr></table> Find the rate of change and the initial value. Explain what each means in this situation.

Hints

- Compare costs for two rows whose hour values differ by one. - Look at the row where the number of hours is zero to identify the initial value.

Solution

1. The cost increases from \(\$6\) to \(\$8\) when time increases from \(0\) to \(1\) hour, so the rate of change is \(\$2\) per hour. 2. When \(h=0\), the cost is \(\$6\), so the initial value is \(\$6\). 3. The garage charges a \(\$6\) starting fee and then \(\$2\) for each hour parked.

Answer

Rate of change: \(\$2\) per hour, the hourly parking charge. Initial value: \(\$6\), the starting fee at \(0\) hours.
Interpret slope and intercept contextually 3
A shuttle bus has \(7\) passengers before its first stop. On a graph with number of stops on the x-axis and number of passengers on the y-axis, what is the y-intercept? What does it mean?

Hints

- What does \(x=0\) mean on this graph? - What does the y-intercept tell you about a starting amount?

Solution

1. Before the first stop, the number of stops is \(0\). 2. The graph has y-intercept \(7\). It means the bus starts with \(7\) passengers.

Answer

The y-intercept is \(7\) passengers. It is the number on the bus before the first stop.
Interpret slope and intercept contextually 4
A dog walker earns \(\$15\) for each hour of work. On a graph of pay in dollars against time in hours, what is the slope? What does it mean?

Hints

- How much more is earned for one more hour? - What units should a rate of pay have?

Solution

1. Each extra hour raises pay by \(\$15\). 2. The slope is \(15\) dollars per hour. It tells how quickly pay increases with hours worked.

Answer

The slope is \(15\,\text{dollars per hour}\). Pay rises by \(\$15\) for every additional hour.
Interpret slope and intercept contextually 5
The graph models the cost \(C\), in dollars, of a ride that travels \(x\) miles. a) Use the graph to find the slope and the y-intercept. b) Interpret both values in the context of the ride.
Figure for problem 551314

Hints

- Use two clearly marked points to compare the change in cost with the change in miles. - The y-intercept is the point where the graph meets the y-axis. - Attach the graph's units to each value when you explain what it means.

Solution

1. Two marked points are \((0,4)\) and \((2,10)\). The slope is \(m=\frac{10-4}{2-0}=\frac{6}{2}=3\), and the line crosses the y-axis at \(4\), so the y-intercept is \(4\). 2. The slope means the cost increases by \(\$3\) for each mile traveled. The y-intercept means the ride has an initial charge of \(\$4\) when \(x=0\).

Answer

a) Slope: \(3\); y-intercept: \(4\). b) The ride costs \(\$3\) per mile plus an initial charge of \(\$4\).
Interpret slope and intercept contextually 6
Two equipment-rental plans charge according to Plan A: \(C = 3h\) Plan B: \(C = 5 + 3h\), where \(C\) is cost in dollars and \(h\) is the number of hours rented. Compare the slopes and y-intercepts of the two models and explain what each comparison means in context.

Hints

- Compare the coefficients of \(h\) first. - Then compare the values of \(C\) when \(h=0\). - Interpret the rate and starting value separately.

Solution

1. Both models have slope \(3\), so both plans charge \(\$3\) per rental hour. 2. Plan A has y-intercept \(0\), so it has no starting fee. 3. Plan B has y-intercept \(5\), so it has a \(\$5\) starting fee. 4. Because the hourly rates are equal, Plan B remains \(\$5\) more expensive for every rental duration.

Answer

Both slopes are \(3\), meaning \(\$3\) per hour. Plan A's y-intercept is \(0\), while Plan B's is \(5\), meaning Plan B has a \(\$5\) starting fee.
Interpret slope and intercept contextually 7
A candle is \(18\,\text{cm}\) tall when it is lit and burns down at a constant rate of \(1.5\,\text{cm}\) per hour. Write a linear equation for its height \(H\), in centimeters, after \(t\) hours. Identify the slope and y-intercept and explain what each means.

Hints

- Decide which quantity is the starting value and which quantity changes each hour. - Use a negative rate because the candle's height decreases. - Put the model in the form initial value plus rate times time.

Solution

1. The starting height is \(18\), so the y-intercept is \(18\). 2. The candle loses \(1.5\) centimeters each hour, so the slope is \(-1.5\). 3. The model is \(H = 18 - 1.5t\).

Answer

\(H = 18 - 1.5t\). The slope is \(-1.5\), meaning the candle loses \(1.5\,\text{cm}\) per hour. The y-intercept is \(18\), the starting height in centimeters.
Interpret slope and intercept contextually 8
One rain barrel contains \(7\) gallons, and its amount of water does not change. A second barrel starts empty and fills at \(5\) gallons per minute. On graphs of water amount against time, which barrel has slope \(0\)? Which has y-intercept \(0\)? Explain each zero.

Hints

- Which barrel's amount changes as time passes? - What does the y-intercept tell you about the amount at the start?

Solution

1. The first barrel's amount is always \(7\) gallons, so its slope is \(0\,\text{gal/min}\): no water is gained or lost. 2. The second barrel has \(0\) gallons at time \(0\), so its y-intercept is \(0\) gallons. Its slope is \(5\,\text{gal/min}\).

Answer

First barrel: slope \(0\,\text{gal/min}\), because its amount stays the same. Second barrel: y-intercept \(0\) gallons, because it starts empty.
Interpret slope and intercept contextually 9
A graph models the percent charge left in a battery. The straight line has been extended below the x-axis. At what time would the charge first reach \(0\%\)? Is the part below \(0\%\) a possible battery charge? Explain.
Figure for problem 560053

Hints

- Where does the line meet the horizontal axis? - What values are possible for the percent charge of a battery? - Can a straight-line rule remain useful outside a realistic time range?

Solution

1. Read the intersection with the x-axis: the charge reaches \(0\%\) at \(5\) hours. 2. A battery cannot have a negative percent charge. The part of the mathematical line below zero is not a valid physical prediction.

Answer

At \(5\) hours. No; negative battery charge is not possible.
Interpret slope and intercept contextually 10
Two monthly cell phone plans have costs, in dollars, modeled by the following expressions. The variable \(m\) is the number of gigabytes of data used. Plan 1: \(10+2m\) Plan 2: \(4m\) a) Describe the base fee and cost per gigabyte for each plan. b) Let \(E\) represent the amount saved by choosing Plan 1 instead of Plan 2. Write and simplify an expression for \(E\). c) Find \(E\) when \(m=3\) and when \(m=7\). Interpret each result.

Hints

- In a cost expression, what does a constant term represent? What does the coefficient of \(m\) represent? - Savings equal the cost not chosen minus the cost chosen. - Use parentheses when subtracting an entire cost expression. - What does a negative savings value mean?

Solution

1. Plan 1 has a base fee of \(\$10\) and costs \(\$2\) per gigabyte. Plan 2 has no base fee and costs \(\$4\) per gigabyte. 2. Savings from choosing Plan 1 are Plan 2 cost minus Plan 1 cost: \(E=4m-(10+2m)=4m-10-2m=2m-10\). 3. When \(m=3\), \(E=2\cdot3-10=-4\), so Plan 1 costs \(\$4\) more. When \(m=7\), \(E=2\cdot7-10=4\), so Plan 1 saves \(\$4\).

Answer

a) Plan 1: \(\$10\) base fee and \(\$2\) per gigabyte. Plan 2: no base fee and \(\$4\) per gigabyte. b) \(E=2m-10\) c) For \(m=3\), \(E=-4\), so Plan 1 costs \(\$4\) more. For \(m=7\), \(E=4\), so Plan 1 saves \(\$4\).
Interpret slope and intercept contextually 11
A delivery service models its total charge by \(C(m)=3m+7\), where \(m\) is the number of miles traveled and \(C(m)\) is the cost in dollars. a) Jordan says that \(3\) is the starting charge and \(7\) is the charge per mile. Casey says that \(7\) is the starting charge and \(3\) is the charge per mile. Who is correct? Explain. b) A second service charges the same amount per mile but has a starting charge that is \(\$2\) higher. Write a linear model for the second service.

Hints

- Decide which part of a linear model remains when the input is zero. - Decide which part tells how much the output changes when the input increases by one unit. - In part b), identify which feature of the original model stays the same and which feature changes.

Solution

1. Casey is correct. In \(C(m)=3m+7\), the coefficient \(3\) is the rate of change, so it represents \(\$3\) per mile. The constant term \(7\) is \(C(0)\), so it represents the \(\$7\) starting charge. 2. The per-mile rate stays \(3\). The new starting charge is \(7+2=9\), so the second service can be modeled by \(S(m)=3m+9\).

Answer

a) Casey is correct: \(\$7\) is the starting charge and \(\$3\) per mile is the rate of change. b) \(S(m)=3m+9\)
Interpret slope and intercept contextually 12
The graph shows water draining from two tanks over the first \(5\) minutes, until the lower tank empties. A student says, “The lower line must catch up to the upper line because both amounts decrease.” Does that happen during these \(5\) minutes? Explain using slopes and starting amounts.
Figure for problem 552010

Hints

- Compare the amounts when time is zero. - How much does each line decrease in one minute? - What happens to the difference when two amounts decrease at the same rate?

Solution

1. The upper line starts at \(14\,\text{gal}\) and the lower line at \(10\,\text{gal}\), a difference of \(4\,\text{gal}\). 2. Both lines have slope \(-2\,\text{gal/min}\). Each tank loses the same amount of water per minute. 3. The difference stays \(4\,\text{gal}\) during the first \(5\) minutes, so the lines do not meet on the graph. The claim is false for this time interval.

Answer

No, not during the first \(5\) minutes. Both amounts decrease by \(2\,\text{gal/min}\), so the \(4\)-gallon difference stays the same while both tanks are draining.
Interpret slope and intercept contextually 13
The graph shows a simplified linear model for air temperature \(T\), in degrees Fahrenheit, at elevation \(e\), measured in thousands of feet. Only elevations from \(2\) to \(8\) thousand feet are shown. Use the marked points when needed. a) Find the slope and interpret it in context. b) Write a linear model relating \(T\) and \(e\). c) Find the y-intercept and explain what it means. Why is that intercept an extrapolated value rather than a value displayed on the graph?
Figure for problem 552011

Hints

- Use the two marked points to find change in temperature divided by change in elevation. - After finding the slope, substitute one graphed point into \(T = me + b\) to determine the intercept. - Compare the input value represented by the y-intercept with the elevations actually shown.

Solution

1. Using \((2, 62)\) and \((4, 54)\), the slope is \(\frac{54-62}{4-2} = -4\). The model predicts a decrease of \(4\) degrees Fahrenheit for each increase of \(1000\) feet in elevation. 2. Write \(T = -4e + b\). Substitute \((2, 62)\): \(62 = -4(2) + b\), so \(b = 70\). Thus \(T = 70 - 4e\). 3. The y-intercept is \(70\), representing the model's predicted temperature at \(e=0\), or sea level. It is extrapolated because the displayed graph begins at \(e=2\), not \(e=0\).

Answer

a) Slope: \(-4\) degrees Fahrenheit per \(1000\) feet; the predicted temperature decreases by \(4\) degrees Fahrenheit for each additional \(1000\) feet of elevation. b) \(T = 70 - 4e\) c) The y-intercept is \(70\) degrees Fahrenheit, the model's predicted sea-level temperature. It is extrapolated because \(e=0\) is outside the displayed range.
Interpret slope and intercept contextually 14
The graph shows a device's battery percentage after it is unplugged. a) Determine the y-intercept and slope and interpret both in context. b) Write a linear model for battery percentage \(B\) after \(t\) hours. c) According to the model, after how many hours will the battery be at \(50\%\)?
Figure for problem 552012

Hints

- Read the battery level where the graph meets the vertical axis. - Use two clear points to find change in percentage divided by change in time. - Use the starting value and rate to write the model before answering part c).

Solution

1. The graph starts at \((0, 90)\), so the y-intercept is \(90\): the battery begins at \(90\%\). 2. From \((0, 90)\) to \((5, 50)\), the change is \(-40\) percentage points over \(5\) hours, so the slope is \(-8\) percentage points per hour. 3. The model is \(B = 90 - 8t\). 4. Set \(B=50\): \(50 = 90 - 8t\). Then \(8t = 40\), so \(t=5\).

Answer

a) y-intercept: \(90\%\), the starting battery level; slope: \(-8\) percentage points per hour. b) \(B = 90 - 8t\) c) \(5\) hours
Interpret slope and intercept contextually 15
A water container starts with \(18\) gallons and loses \(2\) gallons per minute. Write an equation for the water left after \(t\) minutes. Explain the slope and intercept. For how many minutes is this model realistic?

Hints

- Which number describes the amount at time zero? - Should the rate of change be positive or negative? - What happens to the model when the amount of water reaches zero?

Solution

1. Starting amount is \(18\) gallons, so the intercept is \(18\). 2. The amount decreases by \(2\) gallons each minute, so slope is \(-2\) gallons per minute. 3. The equation is \(V=18-2t\). The model applies for \(0\le t\le9\), until the container is empty.

Answer

\(V=18-2t\). The slope is \(-2\) gallons per minute and the intercept is \(18\) gallons. The model applies for \(0\le t\le9\) minutes.

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