Graphing Functions: 4.4

4.4: Graphing Functions

Introduction

This section focuses on the systematic analysis and graphing of polynomial and rational functions. Key concepts include determining the domain, end behavior, intercepts, critical points, and asymptotes. These tools are essential for understanding the qualitative features of function graphs in calculus.

Polynomial Functions

General Properties

• Domain: For polynomials, the domain is typically $(-\infty, \infty)$.

• End Behavior: Determined by the leading term. For $y = ax^n$, as $x \to \pm\infty$, $y \to \pm\infty$ depending on the sign and degree.

• Intercepts:

  • x-intercepts: Solve $y = 0$ for $x$.

  • y-intercept: Set $x = 0$ and solve for $y$.

• Critical Points: Points where $y'$ (the derivative) is zero or undefined. These indicate local maxima, minima, or points of inflection.

Example 1: Quadratic Function $y = x^2 - 4x + 3$

• Domain: $(-\infty, \infty)$

• End Behavior: $\uparrow \quad \uparrow$ (parabola opens upward)

• Intercepts:

  • x-intercepts: $y = x^2 - 4x + 3 = 0 \implies x = 3, 1$ → $(3,0), (1,0)$

  • y-intercept: $x = 0 \implies y = 3$ → $(0,3)$

• Critical Point:

  • Find $y' = 2x - 4 = 0 \implies x = 2$

  • $y(2) = 2^2 - 4(2) + 3 = -1$ → $(2, -1)$

• Graph: Parabola with vertex at $(2, -1)$, intercepts at $(3,0), (1,0), (0,3)$.

Example 2: Cubic Function $y = x^3 - 3x + 3$

• Domain: $(-\infty, \infty)$

• End Behavior: $\downarrow \quad \uparrow$ (as $x \to -\infty$, $y \to -\infty$; as $x \to \infty$, $y \to \infty$)

• Intercepts:

  • x-intercepts: $y = x^3 - 3x + 3 = 0$ (difficult to solve exactly)

  • y-intercept: $x = 0 \implies y = 3$ → $(0,3)$

• Critical Points:

  • $y' = 3x^2 - 3 = 0 \implies x = \pm 1$

  • $y(1) = 1^3 - 3(1) + 3 = 1$ → $(1,1)$

  • $y(-1) = (-1)^3 - 3(-1) + 3 = 5$ → $(-1,5)$

• Graph: Inflection at $x=0$, local extrema at $(1,1)$ and $(-1,5)$.

Example 3: Quartic Function $y = x^4 - 2x^2$

• Domain: $(-\infty, \infty)$

• End Behavior: $\uparrow \quad \uparrow$ (as $x \to \pm\infty$, $y \to \infty$)

• Intercepts:

  • x-intercepts: $x^4 - 2x^2 = 0 \implies x^2(x^2 - 2) = 0 \implies x = 0, \pm\sqrt{2}$ → $(0,0), (\sqrt{2},0), (-\sqrt{2},0)$

  • y-intercept: $x = 0 \implies y = 0$ → $(0,0)$

• Critical Points:

  • $y' = 4x^3 - 4x = 4x(x^2 - 1) = 0 \implies x = 0, \pm 1$

  • $y(0) = 0$ → $(0,0)$

  • $y(1) = 1 - 2 = -1$ → $(1,-1)$

  • $y(-1) = 1 - 2 = -1$ → $(-1,-1)$

• Graph: Double well shape, local minima at $(1,-1)$ and $(-1,-1)$, local maximum at $(0,0)$.

Rational Functions

General Properties

• Domain: Exclude values that make the denominator zero.

• Vertical Asymptotes (V.A.): Occur at values where the denominator is zero and the numerator is nonzero.

• Horizontal Asymptotes (H.A.): Determined by the degrees of numerator and denominator. If degrees are equal, H.A. is the ratio of leading coefficients.

• Intercepts:

  • x-intercepts: Set numerator to zero.

  • y-intercept: Set $x = 0$.

• Critical Points: Where $y'$ is zero or undefined.

Example 4: $y = \dfrac{x^2}{x^2 + 9}$

• Domain: $x^2 + 9 > 0$ for all $x$ → $(-\infty, \infty)$ (no vertical asymptotes)

• Horizontal Asymptote: $\lim_{x \to \pm\infty} \dfrac{x^2}{x^2 + 9} = 1$ → $y = 1$

• Intercepts:

  • x-intercept: $x = 0$ → $(0,0)$

  • y-intercept: $x = 0$ → $(0,0)$

• Critical Point:

  • $y' = \dfrac{18x}{(x^2 + 9)^2}$; $y'(0) = 0$ → $(0,0)$

• Increasing/Decreasing:

  • Increase: $(0, \infty)$

  • Decrease: $(-\infty, 0)$

• Graph: Symmetric about $y$-axis, approaches $y=1$ as $|x| \to \infty$.

Example 5: $y = \dfrac{x+1}{x-2}$

• Domain: $x \neq 2$ (vertical asymptote at $x=2$)

• Vertical Asymptote: $x = 2$

• Horizontal Asymptote: $\lim_{x \to \pm\infty} \dfrac{x+1}{x-2} = 1$ → $y = 1$

• Intercepts:

  • x-intercept: $x+1=0 \implies x=-1$ → $(-1,0)$

  • y-intercept: $x=0 \implies y=-\frac{1}{2}$ → $(0,-\frac{1}{2})$

• Critical Points:

  • $y' = \dfrac{-3}{(x-2)^2}$; never zero, so no critical points.

• Increasing/Decreasing:

  • Increasing: none

  • Decreasing: $(-\infty, 2)$ and $(2, \infty)$

• Graph: Hyperbola with vertical asymptote at $x=2$, horizontal asymptote at $y=1$.

Summary Table: Key Features of Example Functions

Function	Domain	x-intercepts	y-intercept	Critical Points	Asymptotes
$x^2 - 4x + 3$	$(-\infty, \infty)$	$(3,0), (1,0)$	$(0,3)$	$(2,-1)$	None
$x^3 - 3x + 3$	$(-\infty, \infty)$	Hard to solve	$(0,3)$	$(1,1), (-1,5)$	None
$x^4 - 2x^2$	$(-\infty, \infty)$	$(0,0), (\sqrt{2},0), (-\sqrt{2},0)$	$(0,0)$	$(0,0), (1,-1), (-1,-1)$	None
$\dfrac{x^2}{x^2 + 9}$	$(-\infty, \infty)$	$(0,0)$	$(0,0)$	$(0,0)$	H.A.: $y=1$
$\dfrac{x+1}{x-2}$	$x \neq 2$	$(-1,0)$	$(0,-\frac{1}{2})$	None	V.A.: $x=2$, H.A.: $y=1$

Conclusion

Graphing functions involves analyzing their algebraic structure to determine domains, intercepts, critical points, and asymptotes. These features provide a comprehensive understanding of the function's behavior and are foundational for further study in calculus, including optimization and integration.