Comprehensive Guide to Rectangular Coordinate Systems, Graphing Functions, and Intercepts

Rectangular Coordinate System

• Construction of the Rectangular Coordinate System:

  • Formed by drawing a horizontal line and a vertical line that intersect at right angles ($90^\times$).
  • The rectangular coordinate system is the formal mathematical term for a graph.
  • The horizontal line is designated as the $x$-axis.
  • The vertical line is designated as the $y$-axis.
  • The point where the two axes intersect is called the origin, represented by the ordered pair $(0, 0)$.

• Quadrants of the Coordinate Plane:

  • The intersecting axes divide the plane into four regions called quadrants, numbered counterclockwise using Roman numerals:
  • Quadrant I ($\text{Quadrant 1}$): Upper right region. Both coordinates are positive ($+x, +y$).
  • Quadrant II ($\text{Quadrant 2}$): Upper left region. The $x$-coordinate is negative, and the $y$-coordinate is positive ($-x, +y$).
  • Quadrant III ($\text{Quadrant 3}$): Lower left region. Both coordinates are negative ($-x, -y$).
  • Quadrant IV ($\text{Quadrant 4}$): Lower right region. The $x$-coordinate is positive, and the $y$-coordinate is negative ($+x, -y$).

• Ordered Pairs and Coordinates:

  • Each point denoted by a dot in the rectangular coordinate system corresponds to an ordered pair of real numbers $(x, y)$.
  • The order within the pair is mandatory: horizontal positioning always precedes vertical positioning.
  • The $x$-coordinate specifies the horizontal distance and direction from the origin.
  • The $y$-coordinate specifies the vertical distance and direction from the origin.

• Procedure for Plotting Points:

  • Start at the origin $(0, 0)$.
  • Move horizontally along the $x$-axis by the distance and direction specified by $x$ (right for positive, left for negative).
  • Move vertically parallel to the $y$-axis by the distance and direction specified by $y$ (up for positive, down for negative).
  • Plot the dot at the resulting location.

• Plotting Examples and Location Identification:

  • Point A $(2, 6)$: Move right $2$ units, up $6$ units. Located in Quadrant I (both values positive).
  • Point B $(-4, 2)$: Move left $4$ units, up $2$ units. Located in Quadrant II ($x$ negative, $y$ positive).
  • Point C $(0, -5)$: The $0$ in the $x$-slot indicates no horizontal movement from the origin; move down $5$ units. Located on the $y$-axis (not inside any quadrant).
  • Point D \right(-\frac{5}{2}, -\frac{7}{2}\right): Convert fractions to decimals to simplify graphing: $(-2.5, -3.5)$. Move left $2.5$ units, down $3.5$ units. Located in Quadrant III (both values negative).
  • Point E $(4, 0)$: Move right $4$ units, up or down $0$ units. Located on the $x$-axis (not inside any quadrant).
  • Point F $(0, 0)$: Zero horizontal and zero vertical movement. Located at the origin.

Graphs of Common Functions

• Key Parent Functions and Shapes:
  • Linear Function: $f(x) = x$
  • Variable $x$ is to the first power ($x^1$).
  • Graph is a straight diagonal line.
  • Quadratic Function: $f(x) = x^2$
  • Graph is a smooth U-shaped curve (parabola).
  • Cubic Function: $f(x) = x^3$
  • Graph is an S-shaped curve passing through the origin.
  • Absolute Value Function: $f(x) = |x|$
  • Graph forms a V-shape.
  • Square Root Function: $f(x) = \text{square root}(x)$ or $f(x) = \text{sqrt}(x)$
  • Graph starts at $(0,0)$ and curves gradually to the right.
  • Rational Functions:
  • $f(x) = \frac{1}{x}$: Hyperbola split into two curves across opposing quadrants.
  • $f(x) = \frac{1}{x^2}$: Symmetric curves positioned above the horizontal axis.

Graphing Equations via Point-Plotting Method

• Point-Plotting Method Definition:

  • Construct a table (T-chart) of values.
  • Choose a set of $x$-values to substitute into the equation.
  • Evaluate the equation for each $x$-value to calculate the corresponding $y$-value.
  • Form ordered pairs $(x, y)$, plot them on the coordinate grid, and connect them to display the function curve.
  • Graphs can be verified using graphing tools such as Desmos.

• Detailed Example A: Quadratic Function $y = x^2 - 2$

  • Function Type: Quadratic function yielding an upright U-shape curve (since the coefficient of $x^2$ is positive).
  • Evaluation for chosen $x$-values from $-3$ to $3$:
  • For $x = -3$: $y = (-3)^2 - 2 = 9 - 2 = 7 \rightarrow (-3, 7)$. (Parentheses around negative values during squaring are essential to preserve correctness).
  • For $x = -2$: $y = (-2)^2 - 2 = 4 - 2 = 2 \rightarrow (-2, 2)$.
  • For $x = -1$: $y = (-1)^2 - 2 = 1 - 2 = -1 \rightarrow (-1, -1)$.
  • For $x = 0$: $y = 0^2 - 2 = 0 - 2 = -2 \rightarrow (0, -2)$.
  • For $x = 1$: $y = 1^2 - 2 = 1 - 2 = -1 \rightarrow (1, -1)$.
  • For $x = 2$: $y = 2^2 - 2 = 4 - 2 = 2 \rightarrow (2, 2)$.
  • For $x = 3$: $y = 3^2 - 2 = 9 - 2 = 7 \rightarrow (3, 7)$.
  • Symmetry: The output $y$-values show vertical axial symmetry around the vertical axis ($7, 2, -1, -2, -1, 2, 7$).

• Detailed Example B: Absolute Value Function $y = -|x| + 2$

  • Function Type: Absolute value function yielding an inverted (upside down) V-shape due to the negative sign in front of the absolute value expression.
  • Absolute Value Definition: Represents a number's distance away from zero on a number line. Distance is inherently non-negative, so absolute value output is always positive or zero.
  • Order of Operations: Absolute value bars act as grouping symbols (similar to parentheses). Evaluate the expression inside and take its absolute value first before applying external multipliers or negative signs. Distributing a negative into absolute value bars is mathematically invalid.
  • Evaluation for chosen $x$-values from $-3$ to $3$:
  • For $x = -3$: $y = -|-3| + 2 = -(3) + 2 = -1 \rightarrow (-3, -1)$.
  • For $x = -2$: $y = -|-2| + 2 = -(2) + 2 = 0 \rightarrow (-2, 0)$.
  • For $x = -1$: $y = -|-1| + 2 = -(1) + 2 = 1 \rightarrow (-1, 1)$.
  • For $x = 0$: $y = -|0| + 2 = 0 + 2 = 2 \rightarrow (0, 2)$.
  • For $x = 1$: $y = -|1| + 2 = -(1) + 2 = 1 \rightarrow (1, 1)$.
  • For $x = 2$: $y = -|2| + 2 = -(2) + 2 = 0 \rightarrow (2, 0)$.
  • For $x = 3$: $y = -|3| + 2 = -(3) + 2 = -1 \rightarrow (3, -1)$.
  • Plotting: Connecting these coordinates yields an upside-down V-shape centered at vertex $(0, 2)$.

• Detailed Example C: Rational Function $y = \frac{2}{x}$

  • Function Type: Rational function yielding a two-part hyperbola curve (one branch in Quadrant I, one branch in Quadrant III).
  • Simplifying Complex Fractions: Use the Keep, Change, Flip rule when dividing by a fraction:
  • Keep the numerator intact.
  • Change division to multiplication.
  • Flip the denominator fraction to its reciprocal.
  • Multiply straight across top and bottom.
  • Evaluation for chosen $x$-values:
  • For $x = -2$: $y = \frac{2}{-2} = -1 \rightarrow (-2, -1)$.
  • For $x = -1$: $y = \frac{2}{-1} = -2 \rightarrow (-1, -2)$.
  • For $x = -0.5$ ($-\frac{1}{2}$): y = \frac{2}{-\frac{1}{2}} = 2 \times \right(-\frac{2}{1}\right) = -4 \rightarrow (-0.5, -4).
  • For $x = 0.5$ ($rac{1}{2}$): y = \frac{2}{\frac{1}{2}} = 2 \times \right(\frac{2}{1}\right) = 4 \rightarrow (0.5, 4).
  • For $x = 1$: $y = \frac{2}{1} = 2 \rightarrow (1, 2)$.
  • For $x = 2$: $y = \frac{2}{2} = 1 \rightarrow (2, 1)$.

• Detailed Example D: Constant Function $y = 2$

  • General Form: $y = a$, where $a$ is a real constant.
  • Graph Shape: A straight horizontal line.
  • Behavior: Regardless of the input $x$-value selected, the output $y$ is invariant and always equals $2$.
  • Evaluation Table:
  • $x = -3 \rightarrow (-3, 2)$
  • $x = -2 \rightarrow (-2, 2)$
  • $x = -1 \rightarrow (-1, 2)$
  • $x = 0 \rightarrow (0, 2)$
  • $x = 1 \rightarrow (1, 2)$
  • $x = 2 \rightarrow (2, 2)$
  • $x = 3 \rightarrow (3, 2)$

Identifying Intercepts

• $x$--intercept Definitions and Properties:

  • An $x$-intercept is the $x$-coordinate of a point $(x, 0)$ where a graph crosses or touches the $x$-axis.
  • Also referred to as a zero or a root of the function.
  • The $y$-coordinate of an $x$-intercept is always zero ($y = 0$).

• $y$--intercept Definitions and Properties:

  • A $y$-intercept is the $y$-coordinate of a point $(0, y)$ where a graph crosses or touches the $y$-axis.
  • The $x$-coordinate of a $y$-intercept is always zero ($x = 0$).

• Analysis of Example Graphs:

  • Graph A (Linear line graph):
  • Intersects $x$-axis at $-1$: $x$-intercept is $-1$.
  • Intersects $y$-axis at $2$: $y$-intercept is $2$.
  • Graph B (Vertical line graph crossing at $x = 3$):
  • Intersects $x$-axis at $3$: $x$-intercept is $3$.
  • Line runs parallel to the $y$-axis and never crosses it: No $y$-intercept.
  • Graph C (Line passing directly through origin):
  • Intersects both axes at $(0,0)$.
  • $x$-intercept is $0$ and $y$-intercept is $0$ (intercept values can be identical).
  • Graph D (U-shaped/V-shaped symmetric graph):
  • Intersects $x$-axis at two locations: $x$-intercepts are $-2$ and $2$.
  • Intersects $y$-axis down at $-3$: $y$-intercept is $-3$.

Interpreting Graphs and Mathematical Models

• Mathematical Models:

  • Equations or formulas constructed to represent real-world quantitative data trends.

• Model Definitions:

  • $n$: Number of years elapsed since $1990$.
  • $a$: Percentage of high school seniors who had ever used alcohol.
  • $m$: Percentage of high school seniors who had ever used marijuana.
  • Timeframe of data collection model: $1992$ to $2015$.

• Task 1: Graph Estimation:

  • Goal: Estimate the percentage of high school seniors who used alcohol in $2015$ using the graphical line model.
  • Procedure:
  • Locate $2015$ along the horizontal time axis.
  • Trace vertically up to intersect the alcohol data curve.
  • Trace horizontally left to read the value on the vertical percentage axis.
  • Estimation result: The point lies between $40$ and $50$, positioned above the midpoint ($45$) and closer to $50$. Estimated at approximately $47\times$ ($47\text{ percent}$).

• Task 2: Formula Calculation:

  • Model formula for alcohol usage: $a = -0.8n + 62$
  • Determine $n$ for the year $2015$:
  • $n = 2015 - 1990 = 25$
  • Substitute $n = 25$ into the algebraic formula:
  • $a = -0.8(25) + 62$
  • $a = -20 + 62$
  • $a = 42$
  • Formula result: $42\times$ ($42\text{ percent}$).

• Task 3: Model Comparison:

  • Graph estimation yielded $47\times$, whereas the algebraic formula calculated $42\times$.
  • Comparison finding: The mathematical formula model underestimates the percentage compared to the graphical data point.
  • Principle: Mathematical models provide estimations and approximations rather than perfectly exact measurements.

Non-Lecture Dialogue and Room Discussions

• Interaction Summary during Lecture Recording:
  • Door and Key Exchange:
  • Questions raised regarding why the door took long to open and whether keys were brought.
  • Clarification provided that a lecture was actively being recorded.
  • Personal Items and Purchased Supplies Mentioned:
  • Soap holder for soap.
  • Toothbrush holder item.
  • Panty liners.
  • Water bottle / jotting nurse bottle.
  • Toilet bowl cleaning items to prevent staining and control odor.
  • Discussion on Left-handedness:
  • Observations made regarding how left-handed individuals hold pens, grip paper, and tear sheets.
  • Social and Academic Discussions:
  • Discussion regarding sleepovers, school start activities, and room arrangement with roommates.
  • Food references: Pizza rolls, snacks, pizza, and fries.
  • Course communication procedures: Finding professor contact email in syllabus vs. communicating through the Folio system.