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An example of finding the formula of a linear function given two points
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Finding the formula of a linear function
f(x ) = mx + b
f(x) = mx + b
• In August, you used 5 therms of natural gas, and your gas bill was $13.25. In January, you used 38 therms of natural gas and your gas bill was $54.50.
f(x) = mx + b
• In August, you used 5 therms of natural gas, and your gas bill was $13.25. In January, you used 38 therms of natural gas and your gas bill was $54.50.
• Find the equation of the linear function G(t) that determines your monthly gas bill G based on the number of therms of gas t.
G(t) = mt + b
• In August, you used 5 therms of natural gas, and your gas bill was $13.25. In January, you used 38 therms of natural gas and your gas bill was $54.50.
• Find the equation of the linear function G(t) that determines your monthly gas bill G based on the number of therms of gas t.
G(t) = mt + b
• In August, you used 5 therms of natural gas, and your gas bill was $13.25. In January, you used 38 therms of natural gas and your gas bill was $54.50.
• Find the equation of the linear function G(t) that determines your monthly gas bill G based on the number of therms of gas t.
• This means that we have to find a slope and a y-intercept
G(t) = mt + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
G(t) = mt + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• To find the slope, it might be helpful to draw a graph:
G(t) = mt + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• To find the slope, it might be helpful to draw a graph:
G(t) = mt + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• To find the slope, it might be helpful to draw a graph:
therms
charge
G(t) = mt + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• To find the slope, it might be helpful to draw a graph:
therms
charge
5
13.25
G(t) = mt + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• To find the slope, it might be helpful to draw a graph:
therms
charge
5
13.25
38
54.50
f(x) = mx + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• To find the slope, we need a rise
therms
charge
5
13.25
38
54.50
f(x) = mx + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• To find the slope, we need a rise
therms
charge
5
13.25
38
54.50
f(x) = mx + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• To find the slope, we need a rise
therms
charge
5
13.25
38
54.50
41.25
G(t) = mt + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• To find the slope, we need a rise and a run
therms
charge
5
13.25
38
54.50
41.25
G(t) = mt + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• To find the slope, we need a rise and a run
therms
charge
5
13.25
38
54.50
41.25
G(t) = mt + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• To find the slope, we need a rise and a run
therms
charge
5
13.25
38
54.50
41.2533
G(t) = mt + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• The slope is
therms
charge
5
13.25
38
54.50
41.2533
G(t) = mt + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• The slope is which is
therms
charge
5
13.25
38
54.50
41.2533
G(t) = mt + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• The slope is which is = 1.25
therms
charge
5
13.25
38
54.50
41.2533
G(t) = 1.25t + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• We found the slope
G(t) = 1.25t + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• We found the slope, now we need to find the y-intercept
G(t) = 1.25t + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• One way to find the y-intercept is to pick one of the two months
G(t) = 1.25t + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• One way to find the y-intercept is to pick one of the two months (let’s say August)
G(t) = 1.25t + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• One way to find the y-intercept is to pick one of the two months (let’s say August) and plug in the number of therms and the charge.
G(t) = 1.25t + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• One way to find the y-intercept is to pick one of the two months (let’s say August) and plug in the number of therms and the charge.
G(t) = 1.25t + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• One way to find the y-intercept is to pick one of the two months (let’s say August) and plug in the number of therms and the charge.
G(t) = 1.25t + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• One way to find the y-intercept is to pick one of the two months (let’s say August) and plug in the number of therms and the charge.
G(t) = 1.25t + b
G(t) = 1.25t + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• One way to find the y-intercept is to pick one of the two months (let’s say August) and plug in the number of therms and the charge.
13.25 = 1.25∙5 + b
G(t) = 1.25t + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• One way to find the y-intercept is to pick one of the two months (let’s say August) and plug in the number of therms and the charge.
13.25 = 6.25 + b
G(t) = 1.25t + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• One way to find the y-intercept is to pick one of the two months (let’s say August) and plug in the number of therms and the charge.
7 = b
G(t) = 1.25t + b
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• Now we found the y-intercept:
7 = b
G(t) = 1.25t + 7
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• Now we found the y-intercept:
7 = b
G(t) = 1.25t + 7
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• So we have the formula!
G(t) = 1.25t + 7
• August: 5 therms; $13.25• January,: 38 therms; $54.50
• So we have the formula!
G(t) = 1.25t + 7
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