Mathematics
Grade 10
15 min
Graph a two-variable relationship
Graph a two-variable relationship
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1
Introduction & Learning Objectives
Learning Objectives
Identify the independent and dependent variables in a measurement context.
Create a table of values for a two-variable relationship derived from a geometric formula.
Plot ordered pairs from a table of values onto a correctly scaled and labeled Cartesian plane.
Draw a line or curve that accurately represents the relationship between two measurement variables.
Distinguish between linear and non-linear relationships based on their graphical representation.
Interpret the meaning of the slope and intercepts of a graph in the context of a measurement problem.
Use a graph to make reasonable predictions about a measurement relationship.
How does a car's stopping distance change as its speed increases? 🏎️ Graphing the relationship between these two variable...
2
Key Concepts & Vocabulary
TermDefinitionExample
Independent VariableThe variable that is intentionally changed or controlled in a relationship. It is always plotted on the horizontal x-axis.When graphing the area of a square based on its side length, the 'side length' is the independent variable because you choose its value first.
Dependent VariableThe variable that is measured or observed; its value depends on the independent variable. It is always plotted on the vertical y-axis.In the area of a square example, the 'area' is the dependent variable because its value is calculated based on the side length.
Ordered Pair (x, y)A pair of numbers representing a single point on a Cartesian plane. The first value (x) corresponds to the independent variable, and the second value (y) corresponds to the...
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Core Formulas
Slope Formula
m = \frac{y_2 - y_1}{x_2 - x_1}
Calculates the slope (m), or constant rate of change, of a straight line passing through two points, (x₁, y₁) and (x₂, y₂). In measurement, this represents how the dependent variable changes relative to the independent variable.
Linear Equation (Slope-Intercept Form)
y = mx + b
Defines a linear relationship where 'm' is the slope and 'b' is the y-intercept (the value of y when x=0). This is useful for modeling measurement relationships with a constant rate of change and a specific starting value.
Area of a Circle Formula
A = \pi r^2
A common measurement formula that describes a non-linear relationship. Graphing the Area (A) as a function of the radius (r) results in a curve, not a straight line, becaus...
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Challenging
Consider two relationships for a square with side length 's': Perimeter (P = 4s) and Area (A = s²). If both are graphed on the same set of axes (with 's' as the independent variable), which statement is true for s > 4?
A.The perimeter graph is steeper than the area graph.
B.The area graph is a straight line and the perimeter graph is a curve.
C.The value of the perimeter is always greater than the value of the area.
D.The area graph is steeper and its value is greater than the perimeter graph.
Challenging
A graph shows a straight line passing through the origin and the point (5, 12.5). The x-axis represents the side length of a regular pentagon (in cm) and the y-axis represents its perimeter (in cm). What is the geometric relationship (formula) represented by this graph?
A.P = 2.5s
B.P = 5s
C.s = 5P
D.P = 12.5s
Challenging
The volume of a sphere is graphed as a function of its radius (V = (4/3)πr³). How does the slope of the curve change as the radius increases, and what does this signify?
A.The slope is constant, signifying a steady increase in volume.
B.The slope increases, signifying that the volume increases by a larger amount for each 1-unit increase in radius at larger radii.
C.The slope decreases, signifying that the volume is approaching a maximum value.
D.The slope is undefined, as the relationship is a curve.
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