5.1 The Coordinate Plane, Plotting Points & Relations vs. Functions

The Coordinate Plane, Plotting Points & Relations vs. Functions

Quick Summary: The Cartesian coordinate plane is defined by two perpendicular axes intersecting at the origin $(0,0)$, creating four quadrants labeled I through IV in a counterclockwise direction. A relation is any mathematical pairing of inputs ($x$, the domain) with outputs ($y$, the range). A relation qualifies as a function if and only if each input $x$ is mapped to exactly one output $y$. Graphically, this is verified using the Vertical Line Test.

Coordinate geometry bridges pure arithmetic and spatial reasoning. On the HiSET Mathematics subtest, questions routinely require you to locate points, identify domain and range, determine whether a given graph or data set is a function, and evaluate expressions in function notation $f(x)$.

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The Anatomy of the Cartesian Coordinate System

The Cartesian coordinate system is formed by the intersection of two perpendicular number lines on a two-dimensional flat surface (the coordinate plane):

1. The Horizontal Axis ($x$-axis): Values increase to the right (positive) and decrease to the left (negative).
2. The Vertical Axis ($y$-axis): Values increase upward (positive) and decrease downward (negative).
3. The Origin $(0,0)$: The central reference point where the $x$-axis and $y$-axis intersect.

                          y-axis
                            ▲
                            │  Quadrant I
          Quadrant II       │  (+, +)
          (-, +)            │  x > 0, y > 0
          x < 0, y > 0      │
                            │
   ─────────────────────────┼─────────────────────────► x-axis
                            │  Origin (0,0)
          Quadrant III      │  Quadrant IV
          (-, -)            │  (+, -)
          x < 0, y < 0      │  x > 0, y < 0
                            │
                            ▼

Quadrant Properties and Sign Conventions

The four quadrants are conventionally numbered using Roman numerals (I, II, III, IV) in a counterclockwise direction starting from the upper-right quadrant:

Exam Watchout: Points that lie directly on the coordinate axes (such as $(0, 5)$ or $(-3, 0)$) do not belong to any quadrant. If a question asks which quadrant contains $(0, -4)$, the correct answer is that the point lies on the negative $y$-axis, not in Quadrant III or IV.

Plotting Ordered Pairs $(x, y)$

Every point in the coordinate plane is uniquely designated by an ordered pair $(x, y)$:

• The first value, $x$ (the abscissa), indicates how many units to move horizontally from the origin: move right if $x > 0$ and left if $x < 0$.
• The second value, $y$ (the ordinate), indicates how many units to move vertically from the horizontal position: move up if $y > 0$ and down if $y < 0$.

Relations, Domain, and Range

In algebra, any collection of ordered pairs is called a relation. A relation establishes a relationship between two quantities: an input set and an output set.

Defining Domain and Range

• Domain: The set of all possible input values ($x$-coordinates or independent variable values).
• Range: The set of all possible output values ($y$-coordinates or dependent variable values).

                  Relation: {(-2, 4), (0, 1), (3, 7), (5, 4)}
                  
                  Domain (Inputs / x-values):   { -2, 0, 3, 5 }
                  Range (Outputs / y-values):   { 1, 4, 7 }

Note: When listing the domain or range as a mathematical set, duplicate values are written only once, and numbers are typically ordered from least to greatest.

Discrete vs. Continuous Domains

1. Discrete Relations: Composed of individual, separated points (e.g., the number of tickets sold vs. total revenue). The domain and range are listed as explicit sets of distinct numbers: $\text{Domain} = {1, 2, 3, 4}$.
2. Continuous Relations: Formed by an unbroken curve or line segment where all real numbers within an interval are included (e.g., temperature over time). Domain and range are expressed using inequalities or interval notation: $\text{Domain: } -4 \le x \le 6 \quad \text{and} \quad \text{Range: } -2 \le y \le 10$

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The Definition of a Function

A function is a specialized relation with a strict mathematical rule:

$\text{Every input } x \text{ in the domain must map to } \mathbf{\text{EXACTLY ONE}} \text{ output } y \text{ in the range.}$

The "One Output per Input" Principle

To determine whether a relation is a function:

• Look for repeated $x$-values: If an $x$-value appears more than once with different $y$-values, the relation is NOT a function.
• Outputs may repeat: Different $x$-values are permitted to share the same $y$-value. For example, $f(2) = 9$ and $f(-2) = 9$ is completely valid for a function (as seen in parabolas like $y = x^2$).

   VALID FUNCTION (Many-to-One Allowed):     NOT A FUNCTION (One-to-Many Forbidden):
   Input (x)          Output (y)             Input (x)          Output (y)
     -3  ───────────────►  9                   4  ───────────────►  2
      3  ───────────────►  9                   4  ───────────────► -2  (Duplicate x!)
      0  ───────────────►  0                   7  ───────────────►  5

Evaluating Function Representations

The Vertical Line Test (VLT)

The Vertical Line Test is a visual and geometric method used to determine whether a graph on the coordinate plane represents a function.

The Vertical Line Test Rule

If any vertical line can be drawn through a graph such that it intersects the curve at more than one point, the graph is NOT a function.

Why does this rule work? A vertical line represents the equation $x = c$, where the input $x$ is constant. If a vertical line intersects a graph at two points $(c, y_1)$ and $(c, y_2)$ where $y_1 \neq y_2$, then that single input $c$ produces two distinct outputs, violating the fundamental definition of a function.

   PASSES VLT (IS A FUNCTION):               FAILS VLT (NOT A FUNCTION):
          y                                         y
          │      / (Linear)                         │     .---. (Circle)
          │     /                                   │    /     \
     ─────┼────/──────► x                      ─────┼───(───────)───► x
          │   /                                     │    \     /
          │  /                                      │     '---'
          │                                         │       │
     (Every vertical line                      (A vertical line intersects
      hits at most ONCE)                        the circle at TWO points)

Common Graph Classifications on the HiSET

• Valid Functions (Pass VLT): Non-vertical straight lines ($y = mx + b$), upward or downward parabolas ($y = ax^2 + bx + c$), absolute value V-shapes ($y = |x|$), and exponential curves ($y = 2^x$).
• Not Functions (Fail VLT): Circles ($x^2 + y^2 = r^2$), ellipses, sideways parabolas ($x = y^2$), and vertical lines ($x = c$, where the vertical line hits infinitely many points).

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Function Notation: $f(x)$ Mechanics & Evaluation

Function notation replaces the dependent variable $y$ with the symbol $f(x)$, read aloud as "f of x".

$y = f(x)$

• $f$ is the name of the function (other common letters include $g, h, C, P$).
• $x$ represents the input variable (independent variable).
• $f(x)$ represents the resulting output value (dependent variable, equivalent to $y$).

Critical Warning: $f(x)$ does NOT mean $f$ multiplied by $x$. It signifies that the rule named $f$ is being applied to the input value $x$.

Evaluating Functions for Specific Inputs

To evaluate a function for a given input value, substitute that value in place of every instance of the variable $x$ in the algebraic expression.

Step-by-Step Worked Examples

Example 1: Linear Function with a Negative Input

Evaluate $f(-4)$ for the function $f(x) = -3x + 7$.

1. Substitute $-4$ into the expression: $f(-4) = -3(-4) + 7$
2. Apply multiplication of signed numbers: $f(-4) = 12 + 7$
3. Add the terms: $f(-4) = 19$ (This corresponds to the coordinate point $(-4, 19)$ on the graph of $f(x)$).

Example 2: Quadratic Function with Exponent Rules

Evaluate $g(-3)$ for the function $g(x) = 2x^2 - 5x + 3$.

1. Substitute $-3$ for $x$: $g(-3) = 2(-3)^2 - 5(-3) + 3$
2. Evaluate exponents first ($(-3)^2 = +9$): $g(-3) = 2(9) - 5(-3) + 3$
3. Perform multiplications: $g(-3) = 18 + 15 + 3$
4. Sum the values: $g(-3) = 36$

Example 3: Solving for Input When Output is Known

For the function $h(x) = 4x - 9$, find the value of $x$ such that $h(x) = 15$.

1. Set the function expression equal to the target output $15$: $4x - 9 = 15$
2. Add 9 to both sides: $4x = 24$
3. Divide by 4: $x = 6$ (When the input is $6$, the output is $15$, corresponding to the ordered pair $(6, 15)$).

Example 4: Reading Function Values from a Graph

Suppose a graph shows a curve labeled $y = f(x)$:

• To find $f(2)$: Locate $x = 2$ on the horizontal axis, move vertically to meet the curve, and read the corresponding $y$-value on the vertical axis.
• To find $x$ when $f(x) = -3$: Locate $y = -3$ on the vertical axis, move horizontally to find all points where the curve has that height, and read their corresponding $x$-coordinates.