Cartesian Plane, Graph Intercepts, and Algebraic Solutions

The Cartesian Coordinate System

Cartesian Plane Structure:
- The Cartesian plane consists of two perpendicular intersecting line axes used to graph equations and identify precise points: the horizontal $x$ -axis and the vertical $y$ -axis.
- The vertical $y$ -axis runs up and down through the center of the plane.
- The horizontal $x$ -axis runs left and right through the center of the plane.
Directional Sign Conventions:
- On the $x$ -axis:
  - Positive numbers extend to the right of the vertical axis.
  - Negative numbers extend to the left of the vertical axis.
- On the $y$ -axis:
  - Positive numbers extend upward above the horizontal axis.
  - Negative numbers extend downward below the horizontal axis.
Ordered Pair Representation:
- Every point on the Cartesian plane is designated by an ordered pair containing two numbers written in the form $(x, y)$ .
- $x$ -coordinate (First Number): Denotes the horizontal displacement, specifying how many units to move to the right (if positive) or to the left (if negative) from the origin.
- $y$ -coordinate (Second Number): Denotes the vertical displacement, specifying how many units to move up (if positive) or down (if negative) from the origin.
Point Plotting Examples:
- Plotting Point $(1, 2)$ :
  - First number ( $x$ -coordinate) is $1$ : Move $1$ unit to the right.
  - Second number ( $y$ -coordinate) is $2$ : Move $2$ units upward.
  - Location: Positioned in the upper-right region (Quadrant I).
- Plotting Point $(-2, 0)$ :
  - First number ( $x$ -coordinate) is $-2$ : Move $2$ units to the left from the origin.
  - Second number ( $y$ -coordinate) is $0$ : No vertical movement upward or downward.
  - Location: Situated directly on the horizontal $x$ -axis because its $y$ -coordinate is $0$ .
- Plotting Point $(0, -3)$ :
  - First number ( $x$ -coordinate) is $0$ : No horizontal movement to the right or left.
  - Second number ( $y$ -coordinate) is $-3$ : Move $3$ units downward.
  - Location: Situated directly on the vertical $y$ -axis because its $x$ -coordinate is $0$ .

Principles of Graphing and Graph Intercepts

Definition of Graphing an Equation:
- Graphing an algebraic equation represents the collection of all ordered pairs $(x, y)$ that make both sides of the equation equal when substituted for $x$ and $y$ .
- Graphing calculators execute this by substituting a large array of values for $x$ , calculating the corresponding $y$ -values, and plotting all resultant ordered pair points.
Definitions and Mechanics of Intercepts:
- $x$ -intercepts:
  - Definitional point where a graph intersects or touches the $x$ -axis.
  - Because any point on the $x$ -axis has zero vertical displacement, its $y$ -coordinate is always 0$.\n * To find x $-intercepts algebraically without a graphing calculator, set the opposite variable$ y = 0 $in the equation and solve for the$ x-coordinate.\n * **y-intercepts**:\n * Definitional point where a graph intersects or touches the y-axis.\n * Because any point on the y $-axis has zero horizontal displacement, its$ x $-coordinate is always$ 0$.
  - To find $y$ -intercepts algebraically without a graphing calculator, set the opposite variable $x = 0$ in the equation and solve for the $y$ -coordinate.
- General Rule: When solving for a specific intercept, set the other variable to $0$ . Never set the same variable as the desired intercept to 0$.\n\n* **Formatting Requirements**:\n * All intercepts must strictly be written as ordered pairs (x, y) containing parentheses and a comma separating the coordinates.\n * Reporting an intercept as a standalone scalar number (e.g., writing just 3 $instead of$ (3, 0)) is incorrect.\n\n* **Course Material Access**:\n * Annotated lecture slides are published to Canvas using the Microsoft Education OneNote integration.\n * Students with Microsoft OneNote installed can open and view class notes directly in the OneNote application via the Canvas link.\n * Students without OneNote installed can view worked-out slide examples directly on Canvas following each class session.\n\n\n# Step-by-Step Algebraic Solution for Intercepts\n\n* **Target Equation**:\n * y = 4x^2 - 36\n * Note: The term x^2 $(spoken as "$ x $squared" or "$ x $to the power of$ 2 $") denotes$ x $multiplied by itself ($ x \times x).\n\n* **Objective**:\n * Determine all x-intercepts of the given equation algebraically.\n * A graph may possess multiple x-intercepts; every existing intercept must be calculated.\n\n* **Step 1: Substitute y = 0**:\n * To find the x $-intercepts, substitute$ 0 $for the variable$ y:\n * 0 = 4x^2 - 36\n\n* **Step 2: Isolate the Variable x**:\n * Three operations act upon x $: squaring, multiplication by$ 4 $, and subtraction of$ 36. Undo these operations in reverse order.\n * **Add 36 to both sides**:\n * 0 + 36 = 4x^2 - 36 + 36\n * 36 = 4x^2\n * **Divide both sides by 4**:\n * \frac{36}{4} = \frac{4x^2}{4}\n * *Multiplication Table Division Method*: Locate the denominator 4 $in the left margin column, trace across its row to find the numerator$ 36 $, and look up to the top header column to obtain the quotient$ 9$.
  - $\frac{36}{4} = 9$
  - $\frac{4}{4} = 1$
  - Simplified equation: $9 = x^2$
Step 3: Solve for $x$ via Square Root Extraction:
- Taking the square root of both sides undoes the squaring operation.
- Every positive real number possesses two square roots: a positive root and a negative root. Both must be accounted for.
- Positive Root Calculation:
  - $x = \sqrt{9}$
  - $x = 3$
  - Verification: The diagonal entries of a multiplication table ( $1$ , $4$ , $9$ , $16$ ) represent perfect square numbers. Since $3 \times 3 = 3^2 = 9$ , the positive square root of $9$ is 3$.\n * **Negative Root Calculation**:\n * x = -\sqrt{9}\n * x = -3\n * Verification: A negative number multiplied by a negative number yields a positive product ((-3) \times (-3) = 9 $). Thus,$ -3 $is also a valid square root of$ 9$.
Step 4: Formulate Final Ordered Pairs:
- The two solved $x$ -coordinates are $x = 3$ and x = -3$.\n * Combining these with y = 0 $yields two distinct$ x-intercept ordered pairs:\n * (3, 0)\n * (-3, 0)\n\n\n# Questions & Discussion\n\n* **Square Roots of Negative Numbers**:\n * **Student Question**: Can negative numbers have square roots (e.g., half square root)?\n * **Explanation**: Negative numbers do not have real square roots.\n * **Graphical Implication**: If an algebraic step yields an equation such as x^2 = -9 $, there is no real solution for$ x $. Consequently, the graph would have no$ x$$-intercepts.
iClicker Practice Questions:
- In-class iClicker questions are used for conceptual practice and participation, not for formal grading.