4.3 Graphs of Functions, Intercepts, & Symmetry

4.3 Graphs of Functions, Intercepts, & Symmetry

The graph of a function $f$ is the visual representation of all ordered pairs $(x, y)$ in the Cartesian coordinate plane satisfying $y = f(x)$. Graphical analysis allows mathematicians to visualize function behavior, locate critical boundary values, identify structural symmetries, and analyze rates of change without exhaustive point-by-point computation.

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1. Algebraic and Graphical Intercept Determination

Intercepts are the precise points where the graph of a function touches or crosses the coordinate axes.

x-Intercepts (Zeros or Roots)

An x-intercept is a point $(x_i, 0)$ where the graph intersects the horizontal x-axis.

• Algebraic Protocol: Set $y = f(x) = 0$ and solve the resulting algebraic equation for all real values of $x$.
• Multiplicity: A function may have zero, one, or multiple x-intercepts depending on its degree and domain restrictions.
• Real vs. Complex Solutions: Only real solutions to $f(x) = 0$ correspond to visual x-intercepts on the Cartesian plane. Complex roots with non-zero imaginary parts (e.g., $x = 2 \pm 3i$) do not generate x-intercepts.

y-Intercept

A y-intercept is a point $(0, y_0)$ where the graph intersects the vertical y-axis.

• Algebraic Protocol: Evaluate $f(0)$ by substituting $x = 0$ into the function rule.
• Uniqueness: Because a valid function passes the Vertical Line Test, a function can possess at most one y-intercept. If $x = 0$ is outside the function's domain (e.g., $f(x) = \frac{1}{x}$ or $f(x) = \ln(x)$), the graph has no y-intercept.

Worked Example: Complete Intercept Analysis

Find all x-intercepts and y-intercepts of $f(x) = x^3 - 3x^2 - 4x + 12$.

1. y-intercept: Evaluate $f(0)$: $$f(0) = (0)^3 - 3(0)^2 - 4(0) + 12 = 12 \implies (0, 12)$$
2. x-intercepts: Set $f(x) = 0$ and factor by grouping: $$x^3 - 3x^2 - 4x + 12 = 0 \implies x^2(x - 3) - 4(x - 3) = 0$$ $$(x^2 - 4)(x - 3) = 0 \implies (x - 2)(x + 2)(x - 3) = 0$$ Solving gives $x = 2, x = -2, x = 3$. $$\text{x-intercepts: } (-2, 0), (2, 0), \text{ and } (3, 0)$$

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2. Symmetry Analysis: Even and Odd Functions

Symmetry simplifies function analysis by allowing us to deduce the behavior of an entire graph from observing only its right half ($x > 0$).

Types of Function Symmetry

┌─────────────────────────────────────────────────────────┐
│               Geometric Symmetry Pathways               │
├────────────────────────────┬────────────────────────────┤
│   Even Function (y-Axis)   │    Odd Function (Origin)   │
│  - Point (a, b) maps to    │  - Point (a, b) maps to    │
│    (-a, b)                 │    (-a, -b)                │
│  - Left and right sides    │  - 180-degree rotational   │
│    are mirror images       │    point symmetry          │
└────────────────────────────┴────────────────────────────┘

Algebraic Symmetry Testing Protocol

To determine if a function $f(x)$ is even, odd, or neither:

1. Replace every occurrence of $x$ in the formula with $(-x)$ and simplify $f(-x)$.
2. Compare the simplified $f(-x)$ to the original $f(x)$:
   • If $f(-x) = f(x)$, $f$ is even.
   • If $f(-x) = -f(x)$ (every term's sign is negated), $f$ is odd.
   • If neither equality holds, $f$ is neither even nor odd.

Worked Example 1: Rational Function Symmetry Test

Classify $g(x) = \frac{5x^3}{x^2 + 4}$ as even, odd, or neither.

$$\text{Evaluate } g(-x) = \frac{5(-x)^3}{(-x)^2 + 4} = \frac{-5x^3}{x^2 + 4} = -\left(\frac{5x^3}{x^2 + 4}\right) = -g(x)$$

Since $g(-x) = -g(x)$, $g(x)$ is an odd function and possesses origin symmetry.

Worked Example 2: Polynomial Symmetry Test

Classify $h(x) = 2x^4 - 7x^2 + 3$ as even, odd, or neither.

$$h(-x) = 2(-x)^4 - 7(-x)^2 + 3 = 2x^4 - 7x^2 + 3 = h(x)$$

Since $h(-x) = h(x)$, $h(x)$ is an even function and possesses y-axis symmetry.

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3. Monotonic Intervals: Increasing, Decreasing, and Constant Behavior

A function's monotonic behavior describes how the dependent variable $y$ changes as the independent variable $x$ increases from left to right across the domain.

• Increasing on $(a, b)$: A function $f$ is increasing on an open interval $(a, b)$ if for any $x_1 < x_2$ in $(a, b)$, $f(x_1) < f(x_2)$. Graphically, the curve rises from left to right.
• Decreasing on $(a, b)$: A function $f$ is decreasing on an open interval $(a, b)$ if for any $x_1 < x_2$ in $(a, b)$, $f(x_1) > f(x_2)$. Graphically, the curve falls from left to right.
• Constant on $(a, b)$: A function $f$ is constant on an open interval $(a, b)$ if for any $x_1, x_2$ in $(a, b)$, $f(x_1) = f(x_2)$. Graphically, the curve forms a horizontal line segment.

Standard Convention: Intervals of increasing, decreasing, and constant behavior are always written using open interval notation $(a, b)$ referencing the domain's $x$-coordinates.

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4. Relative (Local) Extrema and Turning Points

• Relative (Local) Maximum: A function $f$ has a relative maximum at $x = c$ if there exists an open interval containing $c$ such that $f(c) \ge f(x)$ for all $x$ in that interval. The value $f(c)$ is the relative maximum output value.
• Relative (Local) Minimum: A function $f$ has a relative minimum at $x = c$ if there exists an open interval containing $c$ such that $f(c) \le f(x)$ for all $x$ in that interval. The value $f(c)$ is the relative minimum output value.

Turning Points

Relative extrema occur at turning points where a continuous function changes direction:

• Transition from Increasing to Decreasing $\implies$ Relative Maximum at the peak.
• Transition from Decreasing to Increasing $\implies$ Relative Minimum at the valley.

Worked Example: Quadratic Extrema and Monotonic Analysis

Determine the turning point, extrema, and monotonic intervals for $f(x) = -x^2 + 8x - 12$.

1. Vertex x-coordinate: $x = -\frac{b}{2a} = -\frac{8}{2(-1)} = 4$.
2. Vertex y-coordinate: $f(4) = -(4)^2 + 8(4) - 12 = -16 + 32 - 12 = 4$.
3. Parabola Direction: Since $a = -1 < 0$, the parabola opens downward.
4. Monotonic Intervals:
   • Increasing: $(-\infty, 4)$
   • Decreasing: $(4, \infty)$
5. Relative Extremum: Relative maximum of $4$ occurring at $x = 4$.