Notes on Section 1.2: Visualizing and Graphing Data

One-variable and Two-variable Data

• Definition: One-variable data is a list of values.
  • Example: exam scores: $\{93, 89, 72, 70, 65\}$
• Definition: Two-variable data is a relationship between two data lists.
• Example: The month and the average high temperature by month in Cincinnati.
  • Months: May, June, July, August, September, October
  • Average high temperatures: $74^\circ, 85^\circ, 88^\circ, 85^\circ, 76^\circ, 63^\circ$
  • Expressed as a relation: $R = {(\text{May}, 74^\circ), (\text{June}, 85^\circ), (\text{July}, 88^\circ), (\text{August}, 85^\circ), (\text{September}, 76^\circ), (\text{October}, 63^\circ)}$

The Concept of a Relation: Domain and Range

• A relation is a set of ordered pairs. Notation: an ordered pair is written $(x, y)$.
• The domain is the set of all possible x-values.
• The range is the set of all possible y-values.
• Example: Express the Cincinnati weather data as a relation.
  • Domain: $\text{Domain}(R) = {\text{May}, \text{June}, \text{July}, \text{August}, \text{September}, \text{October}}$
  • Range: $\text{Range}(R) = {74^\circ, 85^\circ, 88^\circ, 85^\circ, 76^\circ, 63^\circ}$
  • Note: Duplicates in a range are typically collapsed in a set, so the distinct range values would be ${74^\circ, 85^\circ, 88^\circ, 76^\circ, 63^\circ}$.
• Additional example: A ball is tossed into the air. The height x seconds after being tossed is given by the table:
  • Seconds: 0, 1, 2, 3, 4
  • Height (feet): 300, 284, 236, 156, 44
  • Express as a relation: $S = {(0, 300), (1, 284), (2, 236), (3, 156), (4, 44)}$
  • Domain: ${0, 1, 2, 3, 4}$
  • Range: ${300, 284, 236, 156, 44}$

Expressing Data as a Relation (practice from the transcript)

• Given a set of data points, express them as a relation using ordered pairs.
• Observations:
  • A relation captures the association between each x-value and its corresponding y-value as pairs $(x, y)$.
  • The domain collects all x-values; the range collects all y-values.

Graphical Representation: The Coordinate Plane and Scatter Plots

• To visualize the relationship, use a coordinate plane (Cartesian Coordinate System).
• The plotted distinct points form a scatter plot.
• Example data points from the ball-height data:
  • $(0, 300), (1, 284), (2, 236), (3, 156), (4, 44)$
• Scatter Plot: a graph with these plotted points on axes; helps identify the nature of the relation (linear, nonlinear, etc.).

Worked Examples

Cincinnati Weather Data

• Data as a relation: $R = {(\text{May}, 74^\circ), (\text{June}, 85^\circ), (\text{July}, 88^\circ), (\text{August}, 85^\circ), (\text{September}, 76^\circ), (\text{October}, 63^\circ)}$
• Domain: ${\text{May}, \text{June}, \text{July}, \text{August}, \text{September}, \text{October}}$
• Range: ${74^\circ, 85^\circ, 88^\circ, 85^\circ, 76^\circ, 63^\circ}$
• Note: duplicates in the range may be shown, but as a set they count distinct values only.

Ball Height Data

• Time (seconds): ${0, 1, 2, 3, 4}$
• Height (feet): ${300, 284, 236, 156, 44}$
• Relation: $S = {(0, 300), (1, 284), (2, 236), (3, 156), (4, 44)}$
• Domain: ${0, 1, 2, 3, 4}$
• Range: ${300, 284, 236, 156, 44}$
• Graphical takeaway: the data can be plotted as a scatter plot on a height vs. time graph.

Car Distance vs Time: A Linear Relation

• Scenario: A car travels at a rate of 65 mph.
• Table:
  • Time (t in hours): ${0, 1, 2, 3, 4, 5}$
  • Distance (d in miles): ${0, 65, 130, 195, 260, 325}$
• Express the data as a relation: $T = {(0, 0), (1, 65), (2, 130), (3, 195), (4, 260), (5, 325)}$
• Domain: ${0, 1, 2, 3, 4, 5}$
• Range: ${0, 65, 130, 195, 260, 325}$
• Graph the relationship: label axes and ordered pairs.
• Observations: The data forms a linear relation (points lie on a straight line), indicating a constant speed.
• Equation (linear relation): $d = 65\,t$
• If expressed as a function of time: $d(t) = 65\,t\quad \text{for } t \in {0,1,2,3,4,5}.$

Graphical Interpretation and Notation

• Key concept: A linear relation has constant slope; for the car data, the slope is $m = \frac{\Delta d}{\Delta t} = 65\ \text{mph}.$
• Scatter plots visualize whether points align on a straight line (linear), curve, or other pattern.
• Distinction (based on the transcript’s emphasis):
  • A relation is any set of ordered pairs.
  • A function is a special kind of relation where each x-value maps to exactly one y-value. The car-distance data is a function of time; the weather data (monthly averages) is also a function (each month has one average high). The ball-height data is likewise a function (each time maps to a single height).

Key Formulas and Notation (summary)

• Ordered pair notation: $(x, y)$
• Relation: $R = {(x<em>1, y</em>1), (x<em>2, y</em>2), \ldots}$
• Domain: $\text{Domain}(R) = {x<em>1, x</em>2, \ldots}\,\text{(the set of all x-values)}$
• Range: $\text{Range}(R) = {y<em>1, y</em>2, \ldots}\,\text{(the set of all y-values)}$
• For the Cincinnati data: $R = {(\text{May}, 74^\circ), (\text{June}, 85^\circ), (\text{July}, 88^\circ), (\text{August}, 85^\circ), (\text{September}, 76^\circ), (\text{October}, 63^\circ)}$
• For ball height: $S = {(0, 300), (1, 284), (2, 236), (3, 156), (4, 44)}$
• For car distance: $T = {(0, 0), (1, 65), (2, 130), (3, 195), (4, 260), (5, 325)}$
• Linear relation example: $d = 65\,t$
• Function form: $$d(t) = 65\,t, \quad t \in {0,1,2,3,4,5}\n

Connections to foundations and real-world relevance

• Visualizing data with coordinate planes helps identify patterns such as linearity, monotonicity, and potential outliers.
• Understanding domain and range clarifies which values are observed and which are possible in a given dataset.
• Real-world interpretations:
  • Cincinnati weather data shows how average highs change across months; domain lists the months, range lists the temperatures observed.
  • Ball height over time demonstrates a physical trajectory where height decreases over time after a toss (with discrete measurements).
  • Car distance vs. time demonstrates constant speed producing a linear relation, easily captured by a linear equation and useful for predicting distance at future times.
• Practical implications: choosing appropriate graph types (scatter plots for relationships, axis labeling, and ordered pairs) aids data interpretation, communication, and decision-making.