1.5: Formulas for Linear Functions

1. Find formulas for the linear functions

[(a)] Passes through the points \((-1,5)\) and \((2,-1)\)

 

 

[(b)] Slope \(2 / 3\) and passes through the point \((5,7)\)

 

 

[(c)] Has \(x\)-intercept 3 and \(y\)-intercept -5

 

 

[(d)] The line intersects the graph of \(f(x)=\dfrac{50}{x^2+1}\) at \(x=1\) and \(x=7\).

 

 

2. Find a formula for the function \(q=f(p)\)

price per bottle p($)	0.5	0.75	1
number of bottles sold q=f(p)	1500	100	500

 

 

 

 

3. Find a formula for the linear function shown on the graph \(s(q)\)

 

4. Find the equations of the lines parallel to and perpendicular to the line \(y+4 x=7\), and through the point \((1,5)\).

 

 

 

5. Table below gives the cost, \(C(n)\), of producing a certain good as a linear function of \(n\), the number of units produced.

\(n\) (units)	100	125	150	175
\(C(n)\)(dollars)	11000	11125	11250	11375

 

a) \(C(175)\)

b) \(C(175)-C(150)\)

c) \(\dfrac{C(175)-C(150)}{175-150}\)

d) Estimate \(C(0)\). What is the economic significance of this value?

e) The fixed cost of production is the cost incurred before any goods are produced. The unit cost is the cost of producing an additional unit. Find a formula for \(C(n)\) in terms of \(n\), given that

Total cost = Fixed cost + Unit cost - Number of units

 

 

 

 

 

6. The demand for gasoline can be modeled as a linear function of price. If the price of gasoline is \(p=\$ 3.10\) per gallon, the quantity demanded in a fixed period is \(q=65\) gallons. If the price rises to \$3.50 per gallon, the quantity demanded falls to 45 gallons in that period.

a) Find a formula for \(q\) in terms of \(p\).

 

b)Explain the economic significance of the slope of your formula.

 

c) Explain the economic significance of the \(q\)-axis and \(p\)-axis intercepts.
 

 

7. In a college meal plan, you pay a membership fee; then 30 meals cost 137.50 dollars and 60 meals cost 235 dollars. Find the cost of the membership fee, and write the formula for the cost of a meal plan, \(C\), in terms of the number of meals, \(n\).
 

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