Plumber Pricing Problem: Linear Function Application

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2015-09-25 23:23

In collections CCSS Support Math II Unit 2

This example demonstrates a newly started plumber using a linear pricing strategy to attract customers, charging $16 per hour for services. The core mathematical concept is the graphical representation and practical application of linear functions (y = mx + b), where $16 per hour represents the slope (unit price) and any fixed fee or minimum charge corresponds to the y-intercept. Construction highlights: Use GeoGebra's Slider to dynamically adjust time parameters, Segment command to draw coordinate axis lines, Polygon to construct area charts showing total costs, If conditional statements to handle calculations for different charge intervals, and TableText to create data tables for intuitive cost visualization. Suitable for middle school to high school mathematics courses, teaching linear function concepts, the practical meaning of slope and intercept, and applications of function graphs in daily life.

GeoGebra Commands & Parameters

Polygon(A, B, C, D) → Toppoly1, a, b, c, d
Polygon(E, F, G, H) → Rightpoly1, e, f, g, h
VerticalText("\text{Cost \textcolor{#000099}{(\textdollar)}") → VertYaxis
LaTeX("\text{Create a table to show how the two quantities described vary.}") → Pg1A
LaTeX("\text{The two quantities described are the number of hours and the total cost of the work.}") → Pg1B
LaTeX((((("\text{\textcolor{#000099}{" * S2) * "} to include the \textcolor{#000099}{\textdollar") * S2) * "} house call fee.}")) → Pg1D
LaTeX((("\text{The total cost can be determined by multiplying the number of hours by \textcolor{#000099}{" * S1) * "} and then adding}")) → Pg1C
LaTeX("\text{Choose several values for the number of hours and calculate the associated cost.}") → Pg1F
LaTeX("\text{Graph the relationship.}") → Pg2A
LaTeX("\text{Use the table of values to graph the relationship.}") → Pg2B
LaTeX("\text{Let \textcolor{#000099}{x} represent the number of hours and \textcolor{#000099}{y} represent the cost in dollars.}") → Pg2C
LaTeX("\text{Determine the slope and what it means in the context of the problem.}") → Pg3A
LaTeX("\text{Subtract.}") → Pg3J
LaTeX("\text{Simplify.}") → Pg3K
LaTeX((("\text{How can the slope be used to determine how many hours of work a customer could get for \textcolor{#000099}{\textdollar" * S3) * "}?}")) → Pg4C
LaTeX((("\text{The number of hours of work that would cost \textcolor{#000099}{\textdollar" * S3) * "} can be estimated from the graphed values.}")) → Pg4D
LaTeX("\text{\textcolor{#000099}{x}-coordinate.}") → Pg4F
Segment(GF, (x(GF), 0)) → i
OrthogonalLine(GF, i) → j
OrthogonalLine(I, j) → k
Intersect(j, k) → J
Segment(GF, J) → l
Vector(G0, GF) → u
Mod(F1, CF1) → Cond
If(Cond ≠ 0, LaTeX[(((("\text{customer could get a full \textcolor{#000099}{" * FF1) * "} hours of work and part of another hour for \textcolor{#000099}{\textdollar") * S3) * "}.}")], LaTeX[(((("\text{customer could get a full \textcolor{#000099}{" * FF1) * "} hours of work for \textcolor{#000099}{\textdollar") * S3) * "}.}")]) → Pg1J
LaTeX((("\text{Standard call fee=\textdollar" * S2) * "}")) → DKT2
Vector(G0, Gtry) → Fnlv
LaTeX((("\text{problem that each hour of work costs \textcolor{#000099}{\textdollar" * SUG) * "}.}")) → Pg4B
LaTeX("\text{solve for \textcolor{#000099}{y}.}") → Pg1Gne
LaTeX("\text{Slope formula}") → Pg3G
LaTeX("\text{slope\textcolor{#000099}{ =\frac{y₂-y₁}{x₂-x₁}}}") → Pg3E
LaTeX((((("\text{slope\textcolor{#000099}{ =\frac{(" * G2) * ")-(") * G1) * ")}{(2)-(1)}}}")) → Pg3F
LaTeX((("\text{slope\textcolor{#000099}{ =\frac{" * SUG) * "}{1}}}")) → Pg3H
LaTeX((("\text{slope\textcolor{#000099}{ =" * SUG) * "}}")) → Pg3L
LaTeX((("\text{determine how many hours of work a customer could get for \textcolor{#000099}{\textdollar" * S3) * "}.}")) → Question3
LaTeX("\text{Determine the slope of the line for jobs lasting from \textcolor{#000099}{2} hours to \textcolor{#000099}{6} hours and}") → Question2
LaTeX((((("\text{Charge per hour:\textcolor{#000099}{\textdollar" * S1) * "}; Standard house call fee:\textcolor{#000099}{\textdollar") * S2) * "}}")) → Question1
LaTeX((((("\text{Therefore, the equation that represents this scenario is \textcolor{#000099}{y = " * S1) * "x + ") * S2) * "}.}")) → Pg1E
TableText({{LaTeX["\text{\mathbf{Number\;of\;hours}}"], LaTeX[(("\text{\textcolor{#000099}{" * H1) * "}}")], LaTeX[(("\text{\textcolor{#000099}{" * H2) * "}}")], LaTeX[(("\text{\textcolor{#000099}{" * H3) * "}}")], LaTeX[(("\text{\textcolor{#000099}{" * H4) * "}}")], LaTeX[(("\text{\textcolor{#000099}{" * H5) * "}}")], LaTeX[(("\text{\textcolor{#000099}{" * H6) * "}}")]}, {LaTeX["\text {\mathbf{Calculation}}"], LaTeX[(((((((("\text{\textcolor{#000099}{" * S1) * "(") * H1) * ") + ") * S2) * " = ") * G1) * " }}")], LaTeX[(((((((("\text{\textcolor{#000099}{" * S1) * "(") * H2) * ") + ") * S2) * " = ") * G2) * " }}")], LaTeX[(((((((("\text{\textcolor{#000099}{" * S1) * "(") * H3) * ") + ") * S2) * " = ") * G3) * " }}")], LaTeX[(((((((("\text{\textcolor{#000099}{" * S1) * "(") * H4) * ") + ") * S2) * " = ") * G4) * " }}")], LaTeX[(((((((("\text{\textcolor{#000099}{" * S1) * "(") * H5) * ") + ") * S2) * " = ") * G5) * "} }")], LaTeX[(((((((("\text{\textcolor{#000099}{" * S1) * "(") * H6) * ") + ") * S2) * " = ") * G6) * "} }")]}, {LaTeX["\text{\mathbf{Cost\, (\textcolor{#000099}{\textdollar})}}"], LaTeX[(("\text{\textcolor{#000099}{" * G1) * "}}")], LaTeX[(("\text{\textcolor{#000099}{" * G2) * "}}")], LaTeX[(("\text{\textcolor{#000099}{" * G3) * "}}")], LaTeX[(("\text{\textcolor{#000099}{" * G4) * "}}")], LaTeX[(("\text{\textcolor{#000099}{" * G5) * "}}")], LaTeX[(("\text{\textcolor{#000099}{" * G6) * "}}")]}}, "vc|_") → Table
LaTeX((("\text{Determine the slope by using the slope formula, slope\textcolor{#000099}{=}\textcolor{#000099}{ \frac{y₂-y₁}{x₂-x₁}}. Let (\textcolor{#000099}{x₁,y₁}) be (\textcolor{#000099}{1},\textcolor{#000099}{" * G1) * "})}")) → Pg3B
LaTeX((("\text{and (\textcolor{#000099}{x₂,y₂}) be (\textcolor{#000099}{2," * G2) * "}). Substitute these values into the slope formula to find the slope of}")) → Pg3C
LaTeX("\text{the line, and then simplify.}") → Pg3D
LaTeX((((("\text{Substitute \textcolor{#000099}{" * G1) * "} for \textcolor{#000099}{y₁}, \textcolor{#000099}{") * G2) * "} for \textcolor{#000099}{y₂}, \textcolor{#000099}{1} for \textcolor{#000099}{x₁}, and \textcolor{#000099}{2} for \textcolor{#000099}{x₂}.}")) → Pg3I
LaTeX((((("\text{The slope is \textcolor{#000099}{" * SUG) * "}, or \textcolor{#000099}{\textdollar") * SUG) * "} per hour. This verifies the given information in the}")) → Pg4A
LaTeX((("\text{Find \textcolor{#000099}{" * S3) * "} on the \textcolor{#000099}{y}-axis, and then look to the right to determine the corresponding }")) → Pg4E
If(Cond ≠ 0, LaTeX[(((("\text{From the graph, it appears that \textcolor{#000099}{\textdollar" * S3) * "} will pay for a little less than \textcolor{#000099}{") * CF1) * "} hours of work.}")], LaTeX[(((("\text{From the graph,it appears that \textcolor{#000099}{\textdollar" * S3) * "} will pay for \textcolor{#000099}{") * CF1) * "} hours of work.}")]) → Pg4G
LaTeX((("\text{Using the rate of \textcolor{#000099}{\textdollar" * S1) * "} per hour, the number of hours can also be found by first subtracting}")) → Pg4H
LaTeX((((((((("\text{\textcolor{#000099}{" * S2) * "} from \textcolor{#000099}{") * S3) * "} and then dividing the result by \textcolor{#000099}{") * S1) * "}. This yields a result of \textcolor{#000099}{") * F1) * "}; therefore, a}")) → Pg4I
LaTeX("\text{\mathbf{slope} the measure of the rate of change of one variable with respect to another}") → WTK1
LaTeX("\text{variable; slope\textcolor{#000099}{ =\frac{y₂-y₁}{x₂-x₁}}\textcolor{#000099}{ =\frac{\triangle y}{\triangle x}}\textcolor{#000099}{=\frac{rise}{run}}; the slope in the equation \textcolor{#000099}{y = mx + b} is \textcolor{#000099}{m}.}") → WTK2
LaTeX((("\text{Charge per hour=\textdollar" * S1) * "}")) → DKT1
LaTeX((("\text{Customer budget = \textdollar" * S3) * "}")) → text1
LaTeX("\text{Let’s use \textcolor{#000099}{1}, \textcolor{#000099}{2}, \textcolor{#000099}{3}, \textcolor{#000099}{4}, \textcolor{#000099}{5}, and \textcolor{#000099}{6}. Substitute each of the values for \textcolor{#000099}{x} in the equation, and then}") → Pg1G