Workbook and Lab Manual - Graphs

Graphical Representation in Diagnostic Ultrasound

• Role of Graphs in Ultrasound: Diagnostic ultrasound relies heavily on graphical representations to communicate physical data and parameters clearly (for example, displaying blood velocity over time).

• Standard Axis Nomenclature:

  • Horizontal Axis ($x$-axis): The line running horizontally from side to side.

  • Vertical Axis ($y$-axis): The line running vertically up and down.

Classification of Variable Relationships

• Unrelated:

  • Definition: Two items or variables that have no association or affiliation with one another.

  • Examples:

    • Hair color and shoe size.

    • Body weight and day of birth.

    • Environmental temperature and day of the week.

• Related or Proportional:

  • Definition: Two items or variables that share an association or affiliation. The exact mathematical form of the relationship does not need to be specified.

  • Examples:

    • Body weight is related to dieting.

    • Santa is related to Christmas.

    • Exam score is related or proportional to studying.

    • Dental health is related to flossing.

• Directly Related or Directly Proportional:

  • Definition: Two items associated such that when the value of one item increases, the value of the other item also increases.

  • Graphical Trajectory: The graph of two directly related variables extends from the lower left to the upper right.

  • Examples:

    • Clothing size is directly proportional to body weight.

    • Age is directly related to experience.

    • Skill is directly proportional to practice.

    • Quality of wine is directly related or directly proportional to age.

  • Graphical Representation (Height vs. Age): When graphing height ($y$-axis) against age ($x$-axis), as the $x$-variable (age) increases, the $y$-variable (height) increases.

• Inversely Related or Inversely Proportional:

  • Definition: Two items associated such that when the value of one item increases, the value of the other item decreases.

  • Graphical Trajectory: The graph of two inversely related variables extends from the upper left to the lower right.

  • Examples:

    • Golf score is inversely related to skill.

    • A car's gas mileage is inversely proportional to its engine size.

    • Grades in school are inversely proportional to partying time.

  • Graphical Representation (Clothing vs. Temperature): When graphing clothing worn ($y$-axis) against temperature ($x$-axis), as the $x$-variable (temperature) increases, the $y$-variable (clothing) decreases.

• Reciprocal Relationship:

  • Definition: A specific type of inverse relationship defined by two numbers whose mathematical product equals $1$. They are inverse because when one number increases, the other decreases.

  • Mathematical Rule: $A \times B = 1$

  • Examples:

    • $2$ and $\frac{1}{2}$ are reciprocals, because $2 \times \frac{1}{2} = 1$.

    • $10$ and $\frac{1}{10}$ are reciprocals, because $10 \times \frac{1}{10} = 1$.

    • Period and frequency are reciprocals.

Units of Measurement and Conversion Principles

• Necessity of Units:

  • All numerical values must be accompanied by an appropriate unit. A standalone number without units is ambiguous and creates uncertainty.

  • Example: The question "How old is Jenny?" cannot be answered comprehensively with just the number "$6$". It must specify whether the age is $6\,\text{hours}$, $6\,\text{days}$, $6\,\text{years}$, or $6\,\text{decades}$.

• Categories of Physical Units:

  • Units of Length, Distance, or Circumference: Measured in linear units, such as centimeters ($\text{cm}$) or feet ($\text{ft}$).

  • Units of Area: Measured in squared linear units, such as square centimeters ($\text{cm}^2$) or square feet ($\text{ft}^2$).

  • Units of Volume: Measured in cubed linear units, such as cubic centimeters ($\text{cm}^3$) or cubic feet ($\text{ft}^3$).

• Mathematical Conventions and Scaling Terminology:

  • Fundamental Correctness: Any fundamentally correct unit is acceptable (for instance, measuring age in seconds is silly but mathematically valid).

  • "Increase by a Factor": Indicates multiplication by the specified number. Increasing by a factor of $6$ means six times larger ($\times 6$).

  • "Decrease by a Factor": Indicates division by the specified number. Decreasing by a factor of $3$ means one-third of the original value ($\div 3$ or $\times \frac{1}{3}$).

  • Percent: A number followed by the word "percent" is dimensionless / unitless.

• Unit Conversion Concepts:

  • Converting units changes the manner of presentation without altering the actual quantitative value ("total picture").

  • Conceptual Equivalencies:

    • $12\,\text{inches} = 1\,\text{foot}$

    • $12\,\text{inches} = \frac{1}{3}\,\text{yard}$

    • Calculating the number of quarters in $1\,\text{dollar}$.

    • Calculating the number of days in $1\,\text{month}$.

  • Conversion Method: Treat conversion fractions as having a value equal to $1$ and carry all unit terms throughout the calculation.

  • Conversion Practice Prompts:

    • Convert $\$5.00$ into dimes (Hint: $\$1 = 10\,\text{dimes}$).

    • Convert $12\,\text{inches}$ into centimeters (Hint: $1\,\text{inch} = 2.54\,\text{cm}$).