Math · College algebra · Worked example
Interpret slope as a rate of change
Water freezes at 0 °C, which is 32 °F, and boils at 100 °C, which is 212 °F. Write the Fahrenheit temperature F as a linear function of the Celsius temperature C, interpret the slope and intercept, and find the temperature where the two scales agree.
Find the slope
From (0, 32) to (100, 212), F rises 180 degrees while C rises 100.
Write the line
The intercept is the Fahrenheit reading at 0 °C, 32, so F = 1.8C + 32. Body temperature, 37 °C, is 98.6 °F.
Interpret the numbers
The slope, 1.8 °F per °C, says each Celsius degree is 1.8 Fahrenheit degrees. The intercept, 32 °F, is the Fahrenheit reading at the freezing point of water.
Find where the scales agree
Set F = C and solve.
Result
F = 1.8C + 32: each Celsius degree is 1.8 Fahrenheit degrees, 0 °C reads 32 °F, and the two scales agree at −40°.
Your turn
A phone plan costs $25 a month plus $0.10 per text. Write the monthly cost for t texts and interpret the slope. How many texts make the bill $40?
Show the answer and explanation
C = 0.1t + 25, with slope $0.10 per text; 150 texts.
The slope is the cost of one more text and the intercept is the $25 base fee. Solving 0.1t + 25 = 40 gives t = 150.
Keep exploring
Graph plots F = 1.8C + 32 with the freezing point, the boiling point and the point where the scales agree.
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