Viewing Windows, Tick Mark Calculations, and Function Representations

Viewing Windows and Scale Mechanics

• Graph Axes and Increment Scale:

  • Tick marks on the $y$-axis and $x$ -axis represent marked increments along the respective axis.

  • The point of intersection between the $x$-axis and $y$-axis is always zero ($0$).

  • $x$-axis Scale: If the $x$-axis tick marks are labeled as $-4, -3, -2, -1, 1, 2, 3, 4, 5$, the consistent increment between consecutive tick marks is $1$. Therefore, the scale on the $x$-axis is $1$.

  • $y$-axis Scale: If the $y$-axis tick marks are labeled as $-8, -6, -4, -2, 2, 4, 6, 8$, the consistent increment between consecutive tick marks is $2$. Therefore, the scale on the $y$-axis is $2$.

• Components of a Viewing Window:

  • A viewing window is defined by the minimum $x$-value ($X_{\text{min}}$), the maximum $x$-value ($X_{\text{max}}$), and the scale on the $x$-axis (increment).

  • Similarly, the $y$-axis viewing window is defined by the minimum $y$-value ($Y_{\text{min}}$), the maximum $y$-value ($Y_{\text{max}}$), and the scale on the $y$-axis.

Calculating the Number of Tick Marks on an Axis

• Formula for the $x$-axis:

  • To find the total number of tick lines on the $x$-axis, subtract the minimum $x$-value ($X_{\text{min}}$, or $a$) from the maximum $x$-value ($X_{\text{max}}$, or $b$), divide the result by the $x$-axis scale, and add $1$: $\text{Number of tick marks on } x\text{-axis} = \frac{X_{\text{max}} - X_{\text{min}}}{\text{Scale}_{x}} + 1$ $\text{Number of tick marks on } x\text{-axis} = \frac{b - a}{\text{scale}} + 1$

• Formula for the $y$-axis:

  • To find the total number of tick lines on the $y$-axis, subtract the minimum $y$-value ($Y_{\text{min}}$, or $d$) from the maximum $y$-value ($Y_{\text{max}}$, or $e$), divide by the $y$-axis scale ($f$), and add $1$: $\text{Number of tick marks on } y\text{-axis} = \frac{Y_{\text{max}} - Y_{\text{min}}}{\text{Scale}_{y}} + 1$ $\text{Number of tick marks on } y\text{-axis} = \frac{e - d}{f} + 1$

• Step-by-Step Procedure:

  1. Identify the maximum value of the axis.

  2. Subtract the minimum value of the axis from the maximum value.

  3. Divide the resulting difference by the axis scale.

  4. Add $1$ to the quotient.

Graphing Verification and Worked Examples

• Worked Example: $x$-axis Tick Marks:

  • Given Values: $X_{\text{max}} = 10$, $X_{\text{min}} = -4$, $\text{Scale}_{x} = 2$.

  • Calculation: $\text{Tick Marks} = \frac{10 - (-4)}{2} + 1$ $\text{Tick Marks} = \frac{10 + 4}{2} + 1$ $\text{Tick Marks} = \frac{14}{2} + 1 = 7 + 1 = 8$

  • Graphical Verification:

  • Point of origin/intersection is $0$.

  • Positive ticks (increment of $2$ up to maximum $10$): $2, 4, 6, 8, 10$

  • Negative ticks (increment of $2$ down to minimum $-4$): $-2, -4$

  • Full sequence of tick locations: $-4, -2, 0, 2, 4, 6, 8, 10$

  • Total tick line count equals $8$, confirming the formula calculation.

• Worked Example: $y$-axis Tick Marks:

  • Given Values: $Y_{\text{max}} = 12$, $Y_{\text{min}} = 0$, $\text{Scale}_{y} = 3$.

  • Calculation: $\text{Tick Marks} = \frac{12 - 0}{3} + 1$ $\text{Tick Marks} = \frac{12}{3} + 1 = 4 + 1 = 5$

  • Graphical Verification:

  • Origin is at $0$ ($Y_{\text{min}} = 0$).

  • Increment by $3$ up to maximum $12$: $0, 3, 6, 9, 12$

  • Total tick line count equals $5$, confirming the formula calculation.

Scatter Plots vs. Line Graphs

• Scatter Plots:

  • Created by plotting individual data points on a coordinate plane without connecting lines between the points.

• Line Graphs:

  • Created by taking a scatter plot and connecting consecutive data points with straight line segments.

• Dataset Example: Monthly Precipitation in Portland, Oregon:

  • Data tracks month number ($x$-values from $1$ to $12$) against average monthly precipitation ($y$-values).

  • Data points listed: $(1, 6.2)$, $(2, 3.9)$, $(3, 6.2)$, $(4, 3.9)$, $(5, 1.1)$, $(6, 1.6)$, $(7, 3.1)$, $(8, 5.2)$, $(9, 6.4)$, $(10, 3.1)$, $(11, 5.2)$, $(12, 6.4)$.

Using the Desmos Graphing Calculator

• Entering Data into Desmos:

  1. Open Desmos Graphing Calculator.

  2. Click the plus sign ($+$, top left corner).

  3. Select Table from the dropdown menu (options include Expression, Notes, Table).

  4. Populate column $X_1$ with independent variables (e.g., months $1$ through $12$).

  5. Populate column $Y_1$ with dependent variables (e.g., precipitation values).

  6. Desmos automatically generates the scatter plot points upon data entry.

• Viewing and Navigating Points:

  • Zoom out by clicking the minus symbol ($-$) if data points extend beyond default view.

  • Clicking any individual point on the graph reveals its specific $(x, y)$ coordinates (e.g., $(1, 6.2)$).

• Adjusting the Viewing Window in Desmos:

  • Open the settings panel via the tool/wrench icon at the top right corner.

  • $x$-axis Adjustments:

  • Default minimum $x$-value often starts at $-20$.

  • Set $X_{\text{min}}$ to $0$ or $1$ (at or slightly before first data value).

  • Set $X_{\text{max}}$ to $12$ or $13$ (at or slightly past maximum data value).

  • Set Step (scale/increment) to $1$.

  • Label axis as months.

  • $y$-axis Adjustments:

  • Set $Y_{\text{min}}$ according to lowest data value (e.g., $0.5$).

  • Set $Y_{\text{max}}$ according to highest data value (e.g., $6.4$).

  • Label axis as precipitation.

  • Select an appropriate step value or allow Desmos to assign one automatically.

• Converting Scatter Plot to Line Graph in Desmos:

  1. Click and hold the table symbol located above the $Y_1$ column header.

  2. Toggle the Lines option to ON.

  3. Choose the line style: continuous (solid) line or dotted line.

Graphing Calculators for Exams

• Exam Device Policies:

  • Computers, mobile phones, and web apps (such as Desmos) are strictly prohibited during formal examinations.

  • Students must use a dedicated physical graphing calculator.

• Recommended Calculator Models:

  • Texas Instruments TI-83 or TI-84.

  • TI-84 is the primary recommended model.

  • Calculators can be borrowed from the institutional library if necessary.

• Standard Default Calculator Window Settings:

  • $X_{\text{min}} = -10$

  • $X_{\text{max}} = 10$

  • Scale/Increment = $1$ on both axes.

Section 1.3: Functions and Their Representations

• Four Methods of Representing Functions:

  1. Verbal: Described using words.

  2. Symbolic / Equations: Expressed using mathematical symbols and equations.

  3. Graphical: Visualized by plotting a graph of the function on a coordinate plane.

  4. Numerical: Represented using data tables.

• Domain and Range Concepts:

  • Domain: The set of all valid $x$-values; represents all inputs of the function.

  • Range: The set of all valid $y$-values; represents all outputs of the function.

  • Expressed using set-builder notation or interval notation.

Definition and Characteristics of Functions

• Mathematical Definition of a Function:

  • A process or computation that receives inputs and produces outputs.

• Core Characteristic of Functions:

  • For each particular input, a function must always produce exactly one unique output.

• Testing for a Function:

  • Select an input value, substitute it into the relation, and evaluate.

  • If the input yields exactly one output value, the relation is a function.

  • If an input yields more than one output value, the relation is not a function.

• Real-World Application: Distance of Lightning:

  • Data models approximate distance $y$ (in miles) to a lightning bolt given a time lapse of $x$ seconds between seeing lightning and hearing thunder.

  • Rule: $y$ is calculated by dividing $x$ by $5$ ($y = \frac{x}{5}$).

  • Input $x = 5 \implies y = 1$

  • Input $x = 10 \implies y = 2$

  • Input $x = 15 \implies y = 3$

  • Input $x = 20 \implies y = 4$

  • Input $x = 25 \implies y = 5$

  • Because every unique input $x$ produces exactly one output $y$, this relationship defines a valid function.

• Counterexample (Non-Function Table):

  • If a table contains repeating input values with differing outputs:

  • Input $x = 5 \implies y = 1$

  • Input $x = 10 \implies y = 2$

  • Input $x = 15 \implies y = 3$

  • Input $x = 5 \implies y = 2$

  • The single input value $5$ maps to two distinct outputs ($1$ and $2$).

  • This violates the core rule of functions; therefore, it does not qualify as a function.

Functional Notation

• Function Designation:

  • Functions are commonly designated using the letter $f$ (or $g$, $h$, etc.).

• Notation Expressions:

  • Statement: "$y$ is a function of $x$".

  • Mathematical notation: $y = f(x)$ $y = g(x)$

  • In this notation, $x$ represents the input (domain element) and $f(x)$ represents the output (range element, $y$).