Graphing Related Quantities

Many real-world situations involve two quantities that change together — the height of water in a tank as it fills, the distance a car has traveled over time, or the area of a rectangle as one side grows. Graphing related quantities means translating a verbal or physical description into a coordinate graph where one axis represents an input quantity and the other represents an output quantity that depends on it. This skill bridges the gap between abstract algebra and the concrete world. When we graph related quantities, we are really building a visual model of a function. The shape of the graph — whether it rises steeply, levels off, curves upward, or decreases — encodes information about the rate of change and the nature of the relationship. Polynomial and rational functions appear frequently in these contexts: a falling object's height is quadratic in time, the intensity of light from a source is a rational function of distance, and the volume of a container can be a cubic polynomial in a length measurement. Constructing these graphs requires careful attention to domain (what input values make physical sense), range (what output values are actually achievable), intercepts (where the quantity equals zero or starts), and concavity (whether the rate of change itself is increasing or decreasing). By reading a context critically, labeling axes with correct units, and sketching an accurate curve, students develop a powerful tool for analysis that goes far beyond plotting points by hand.

Key takeaways
  • Identify the Two Quantities
  • Determine the Realistic Domain
  • Write the Functional Relationship
  • Find Key Features: Intercepts, Extrema, and Concavity
  • Scale the Axes and Plot
  • Interpret the Graph in Context

Identify the Two Quantities

Read the problem and name the independent variable (input, usually along the x-axis) and the dependent variable (output, usually along the y-axis). Decide what units each carries. For example, if water flows into a cone-shaped tank, the independent variable might be time t (seconds) and the dependent variable might be water depth h (centimeters).

Determine the Realistic Domain

The mathematical domain of a function may be all real numbers, but the contextual domain is restricted to values that make physical sense. If a box is being built from a sheet of cardboard that is 20 cm wide, the cut length x satisfies 0 < x < 10 (half the width), so the domain is the open interval (0, 10). Always state the domain in context.

Write the Functional Relationship

Express the dependent variable as a formula in terms of the independent variable. Use geometric formulas, physics laws, or algebraic reasoning. For the open box example with a 20 cm × 30 cm sheet and square corner cuts of side x: Volume V(x) = x(20 − 2x)(30 − 2x) = 4x³ − 100x² + 600x. This is a cubic polynomial.

Find Key Features: Intercepts, Extrema, and Concavity

Set V(x) = 0 to find x-intercepts (where volume is zero): x = 0, x = 10, x = 15 — but only x = 0 and x = 10 lie in the domain. Compute V'(x) = 12x² − 200x + 600 and solve V'(x) = 0 to find critical points. The discriminant gives x = (200 ± √(40000 − 28800))/24 = (200 ± √11200)/24 ≈ 3.92 cm as the maximum in (0, 10). Also evaluate concavity via V''(x) = 24x − 200.

Scale the Axes and Plot

Choose axis scales that fit the domain and range. Mark the key points: the intercepts, the maximum (≈ 3.92, 1056.6), and a few check points such as V(2) = 4(8) − 100(4) + 600(2) = 32 − 400 + 1200 = 832 and V(8) = 4(512) − 100(64) + 600(8) = 2048 − 6400 + 4800 = 448. Draw a smooth curve through these points, matching the concavity: concave down near x ≈ 3.92 (local max) and concave up near x close to 0 or 10.

Interpret the Graph in Context

After drawing the graph, extract meaning. The local maximum tells you the cut size that produces the largest box. The x-intercepts show where the box has zero volume (degenerate cases). The shape of the curve shows that increasing x too much beyond the optimal value rapidly decreases volume. Always tie every graphical feature back to a real physical meaning.

The Key Equation

V(x)=x(L−2x)(W−2x)V(x) = x(L - 2x)(W - 2x)
  • V(x) — Volume (cm³ (or appropriate cubic unit)): The volume of the open box as a function of cut length x.
  • x — Cut length (cm): The side length of each square cut from the corners; also the height of the resulting box.
  • L — Sheet length (cm): The longer dimension of the original rectangular sheet of material.
  • W — Sheet width (cm): The shorter dimension of the original rectangular sheet of material.

Worked Example

Problem. A 12 cm × 18 cm piece of cardboard has equal squares of side x cut from each corner, and the sides are folded up to form an open box. (a) Write V(x), the volume as a polynomial in x. (b) State the realistic domain. (c) Find the value of x that maximizes volume, accurate to two decimal places.

x=5−7≈2.35 cm,Vmax⁡≈228.2 cm3x = 5 - \sqrt{7} \approx 2.35 \text{ cm}, \quad V_{\max} \approx 228.2 \text{ cm}^3

Answer. The cut length that maximizes volume is x = 5 − √7 ≈ 2.35 cm, giving a maximum volume of approximately 228.2 cm³.

Figure
Diagram
V(x) = 4x³ − 60x² + 216x on (0, 6)xV012345650100150200max ≈ (2.35, 228)x = cut length (cm)Volume (cm³)
Origins · 1 minRené Descartes and how this was discovered
RD
René Descartes
1596–1650 · French

Descartes invented the Cartesian coordinate system — the x-y plane — which is the very foundation of graphing related quantities. By assigning a pair of numbers (x, y) to every point in the plane, he created a bridge between algebra and geometry that made it possible to represent any functional relationship as a visible curve. His 1637 work La Géométrie introduced algebraic notation for curves and laid the groundwork for analytic geometry and, later, calculus. Descartes reportedly developed many of his philosophical and mathematical ideas while lying in bed in the morning, a habit he maintained for most of his adult life. He only abandoned this routine when Queen Christina of Sweden demanded 5 a.m. philosophy lessons — some historians believe the cold Stockholm mornings contributed to the pneumonia that killed him.

  • Domain and Range of Functions — Every graph of related quantities requires identifying a realistic domain (valid inputs) and range (achievable outputs), making domain and range fundamental prerequisites.
  • End Behavior of Polynomials — The shape and direction of polynomial graphs in context — whether volume grows or collapses near the boundary of the domain — is governed by end behavior of the polynomial function.
  • Optimization with Polynomials — Finding the maximum or minimum of a quantity graphed in context (such as maximum box volume) is the central goal of polynomial optimization problems.
  • Average Rate of Change — The steepness and direction of a contextual graph is described by rates of change, connecting the visual graph to numerical measures of how fast one quantity changes relative to another.