Cartesian Coordinate Systems, Linear Equations, and Graphing Methods

Fundamentals of the Cartesian Coordinate System

• Terminology and Alternative Names:

  • The system used to locate specific points on a graph is formally known as the Cartesian coordinate system or the rectangular coordinate system. These terms are used interchangeably.

• Axes and Central Reference Point:

  • x-axis: The horizontal axis on the coordinate grid.
  • y-axis: The vertical axis on the coordinate grid.
  • Origin: The exact center point where the x-axis and y-axis intersect. Its specific ordered pair location is $(0, 0)$. The first zero indicates no horizontal movement, and the second zero indicates no vertical movement.
  • Coordinate positioning functions on absolute location principles, meaning point evaluation always begins at the origin $(0, 0)$.

• Quadrants:

  • The intersection of the horizontal x-axis and vertical y-axis divides the plane into four distinct contiguous region blocks known as quadrants.
  • Quadrant I: Located in the upper right-hand region of the coordinate plane.
  • The remaining quadrants are numbered sequentially in a counterclockwise direction starting from Quadrant I:
  • Quadrant I: Upper right corner ($+x, +y$).
  • Quadrant II: Upper left corner ($-x, +y$).
  • Quadrant III: Lower left corner ($-x, -y$).
  • Quadrant IV: Lower right corner ($+x, -y$).

Ordered Pairs and Quadrant Movement

• Definition and Structure:

  • Locations on a coordinate plane are specified using an ordered pair, formatted strictly as $(x, y)$.
  • Component order is mandatory: the first value represents the $x$-coordinate and the second value represents the $y$-coordinate.
  • Global positioning systems (GPS), longitude, and latitude operate on this identical ordered pair principle.

• Movement Rules for the First Value ($x$-coordinate):

  • Movement always begins at the origin $(0, 0)$:
  • If $x > 0$ (positive): Move to the right by the given number of units.
  • If $x < 0$ (negative): Move to the left by the given number of units.
  • If $x = 0$: Stay put horizontally.

• Movement Rules for the Second Value ($y$-coordinate):

  • Executed sequentially immediately following the horizontal $x$-movement:
  • If $y > 0$ (positive): Move up by the given number of units.
  • If $y < 0$ (negative): Move down by the given number of units.
  • If $y = 0$: Stay put vertically.

• Step-by-Step Point Plotting Examples:

  • $(2, 1)$: From the origin, move right $2$ units, then move up $1$ unit.
  • $(-3, 1)$: From the origin, move left $3$ units, then move up $1$ unit.
  • $(0, 4)$: From the origin, stay put horizontally, then move up $4$ units.
  • $(0, -5)$: From the origin, stay put horizontally, then move down $5$ units.
  • $(5, 0)$: From the origin, move right $5$ units, then stay put vertically.
  • $(-5, -6)$: From the origin, move left $5$ units, then move down $6$ units.
  • $(4, -3)$: From the origin, move right $4$ units, then move down $3$ units.
  • $(-6, 5)$: From the origin, move left $6$ units, then move up $5$ units.
  • $(0, 0)$: Stay put horizontally and stay put vertically; represents the exact location of the origin.

Linear Equations and Graph Solutions

• Geometric Principles of Lines:

  • By geometric definition, a straight line extends infinitely in both opposite directions.
  • A straight line contains an infinite number of distinct points.
  • A minimum of two points is geometrically required to establish a straight line.

• Algebraic Solution vs. Visual Graph Connection:

  • Linear equations represent two-variable relations whose visual output is a straight line.
  • Solution to a Linear Equation ($\text{soln}$): Any ordered pair $(x, y)$ that lies directly on the drawn line is an algebraic solution. Substituting these coordinates into the equation yields a balanced, true equality.
  • Any point that does not touch or lie on the graphed line does not balance the equation scale and is therefore not a solution.

• Testing Solutions Algebraically:

  • To test whether an ordered pair is a solution without a visual graph, substitute the x-value and y-value into the equation using hollowed-out parentheses to represent the substitution.

• Test Case 1: Is $(6, 0)$ a solution to $x - 2y = 6$?

  • Substitute $x = 6$ and $y = 0$:     $(6) - 2(0) = 6$
  • Simplify left side:     $6 - 0 = 6$$6 = 6$
  • Conclusion: The scale is balanced ($\text{True}$). Yes, $(6, 0)$ is a solution to the equation and touches the line.

• Test Case 2: Is $(2, 0)$ a solution to $x = -\frac{1}{3}y$?

  • Substitute $x = 2$ and $y = 0$:     $(2) = -\frac{1}{3}(0)$
  • Simplify right side:     $2 = 0$
  • Conclusion: The scale is unbalanced ($\text{False}$). No, $(2, 0)$ is not a solution and does not touch the line.

Intercept Methods for Graphing Linear Equations

• Definition of Intercepts:

  • $x$-intercept: The specific location where a line touches or crosses the horizontal $x$-axis.
  • $y$-intercept: The specific location where a line touches or crosses the vertical $y$-axis.

• Blueprint Coordinates for Intercepts:

  • $x$-intercept Blueprint: $(\text{number}, 0)$ or $(x, 0)$. The $y$-value is forced to be $0$.
  • $y$-intercept Blueprint: $(0, \text{number})$ or $(0, y)$. The $x$-value is forced to be $0$.
  • Rule: When looking at an intercept coordinate, the variable that equals zero identifies the opposite variable as the intercept axis.

• Procedure to Calculate Intercepts Algebraically:

  • Finding the $x$-intercept: Force $y = 0$ in the linear equation and solve for $x$. Express as an ordered pair $(x, 0)$.
  • Finding the $y$-intercept: Force $x = 0$ in the linear equation and solve for $y$. Express as an ordered pair $(0, y)$.

• Comprehensive Intercept Example 1: Graph $3x - 5y = -15$ using intercepts.

  • Calculate $x$-intercept:
  • Force $y = 0$:       $3x - 5(0) = -15$$3x = -15$
  • Divide both sides by $3$:       $\frac{3x}{3} = \frac{-15}{3}$$x = -5$
  • Ordered Pair Location: $(-5, 0)$.
  • Calculate $y$-intercept:
  • Force $x = 0$:       $3(0) - 5y = -15$$-5y = -15$
  • Divide both sides by $-5$:       $\frac{-5y}{-5} = \frac{-15}{-5}$$y = 3$
  • Ordered Pair Location: $(0, 3)$.
  • Plotting: Plot $(-5, 0)$ and $(0, 3)$ on the grid and connect them to form the line.

• Comprehensive Intercept Example 2: Graph $6x + y = 12$ using intercepts.

  • Calculate $y$-intercept:
  • Force $x = 0$:       $6(0) + y = 12$$y = 12$
  • Ordered Pair Location: $(0, 12)$.
  • Calculate $x$-intercept:
  • Force $y = 0$:       $6x + (0) = 12$$6x = 12$
  • Divide both sides by $6$:       $\frac{6x}{6} = \frac{12}{6}$$x = 2$
  • Ordered Pair Location: $(2, 0)$.

• Digital Homework Input Rules (MyMathLab):

  • Answers must be entered as full ordered pairs formatted with parentheses $(x, y)$, not single scalar values.
  • Homework problem directions dictate required methods:
  • "Find the intercepts and then graph" requires explicit calculation and entry of intercept ordered pairs.
  • "Graph" allows the use of any valid graphing technique.

Slope-Intercept Form and Calculator-Based Graphing

• Slope-Intercept Form:

  • Standard form: $y = mx + b$
  • $m$ represents slope (defined as $\frac{\text{rise}}{\text{run}}$).
  • $b$ represents the $y$-intercept, giving point $(0, b)$.
  • Example: $y = 2x + 5$
  • $y$-intercept is $(0, 5)$.
  • Slope $m = 2 = \frac{2}{1}$ (rise $2$ units up, run $1$ unit right).
  • Starting at $(0, 5)$, rise $2$ and run right $1$ to locate the second point $(1, 7)$, then draw the line.

• Calculator-Ready Form:

  • Before graphing an equation on a handheld graphing calculator, the equation must be isolated for $y$.

• TI-Graphing Calculator Steps:

  1. Solve the equation for $y$.
  2. Press Y= to open the equation editor.
  3. Input the expression using the variable key X,T,\theta,n for $x$.
  4. Press GRAPH to view the plot.
  5. Press 2nd then GRAPH (TABLE) to display the x-y coordinate table.
  6. Select "clean coordinates" (ordered pairs with non-decimal integer values) from the table to plot points manually or digitally.

• Standard Viewing Window Settings:

  • Default window dimensions are $10 \times 10$ ($-10$ to $10$ along both $x$ and $y$ axes).
  • To reset to standard window settings, press ZOOM and select option 6 (Zoom Standard).

• Comprehensive Calculator/Table Example: Graph $9x - 4y = 16$

  • Convert to Calculator-Ready Form:
  • Subtract $9x$ from both sides:       $-4y = -9x + 16$
  • Divide every term by $-4$:       $\frac{-4y}{-4} = \frac{-9x}{-4} + \frac{16}{-4}$$y = \frac{9}{4}x - 4$
  • Calculator Input:
  • Enter (9 / 4) * X - 4 into Y=, using explicit parentheses around the fraction term (9 / 4).
  • Table Extraction (2nd + TABLE):
  • Clean Coordinate 1: $(0, -4)$
  • Clean Coordinate 2: $(4, 5)$
  • Plotting: Mark $(0, -4)$ and $(4, 5)$ on the coordinate plane to complete the line.

Classroom Identification Exercises

• Identified Graph Coordinates:
  • Point A: $(-3, 5)$
  • Point B: $(5, 1)$
  • Point C: $(-4, -3)$
  • Point D: $(2, -4)$
  • Point F: $(-2, 0)$