Sketching Curves: Intercepts, Asymptotes, and Critical Points in Business Calculus

Sketching a Curve: Intercepts, Asymptotes, and Critical Points

Introduction

In Business Calculus, understanding how to sketch the graph of a function is essential for visualizing its behavior and analyzing key features such as intercepts, asymptotes, and critical points. This section focuses on applying calculus techniques to graph polynomial functions, identifying important points, and interpreting their significance.

Graphing Polynomial Functions

Example A: Sketching f(x) = x^2 + 5x - 6

This example demonstrates how to sketch a quadratic function by finding its intercepts, critical points, and analyzing its curvature.

• Function: $f(x) = x^2 + 5x - 6$

• First Derivative (Slope): $f'(x) = 2x + 5$

  • Set $f'(x) = 0$ to find critical points: $2x + 5 = 0 \Rightarrow x = -\frac{5}{2}$

• Second Derivative (Concavity): $f''(x) = 2$

  • Since $f''(x)$ is positive, the graph is concave up everywhere.

• Relative Minimum: Occurs at $x = -\frac{5}{2}$

  • Find $f(-\frac{5}{2})$: $f(-\frac{5}{2}) = (-\frac{5}{2})^2 + 5(-\frac{5}{2}) - 6 = \frac{25}{4} - \frac{25}{2} - 6 = -12.25$

  • Relative minimum point: $(-2.5, -12.25)$

• Y-intercept: Set $x = 0$

  • $f(0) = 0^2 + 5 \times 0 - 6 = -6$

  • Y-intercept: $(0, -6)$

• X-intercepts: Set $f(x) = 0$

  • $x^2 + 5x - 6 = 0$

  • Factor: $(x - 1)(x + 6) = 0$

  • Solutions: $x = 1$, $x = -6$

  • X-intercepts: $(1, 0)$ and $(-6, 0)$

• Graph Shape: Parabola opening upwards due to positive leading coefficient.

Example: The graph passes through $(0, -6)$, has x-intercepts at $(1, 0)$ and $(-6, 0)$, and a relative minimum at $(-2.5, -12.25)$.

Example B: Sketching f(x) = (1 - 2x)^3

This example illustrates how to sketch a cubic function, identify its intercepts, and analyze its critical points.

• Function: $f(x) = (1 - 2x)^3$

• First Derivative: $f'(x) = 3(1 - 2x)^2 \cdot (-2) = -6(1 - 2x)^2$

  • Set $f'(x) = 0$ to find critical points: $-6(1 - 2x)^2 = 0 \Rightarrow 1 - 2x = 0 \Rightarrow x = \frac{1}{2}$

• Second Derivative: $f''(x) = -12(1 - 2x)(-2) = 24(1 - 2x)$

  • At $x = \frac{1}{2}$, $f''(\frac{1}{2}) = 0$ (test for inflection point)

• Critical Point: $(\frac{1}{2}, 0)$

  • Check if it is a minimum, maximum, or neither using the first and second derivative tests.

  • Since $f''(\frac{1}{2}) = 0$, use the first derivative test: $f'(x) \leq 0$ indicates neither a min nor max.

• Y-intercept: Set $x = 0$

  • $f(0) = (1 - 2 \times 0)^3 = 1^3 = 1$

  • Y-intercept: $(0, 1)$

• X-intercept: Set $f(x) = 0$

  • $(1 - 2x)^3 = 0 \Rightarrow 1 - 2x = 0 \Rightarrow x = \frac{1}{2}$

  • X-intercept: $(\frac{1}{2}, 0)$

• Graph Shape: Cubic curve with a point of inflection at $x = \frac{1}{2}$.

Example: The graph passes through $(0, 1)$, has an x-intercept at $(\frac{1}{2}, 0)$, and a critical point at $(\frac{1}{2}, 0)$ which is an inflection point.

Summary Table: Key Features of Example Functions

Function	Y-intercept	X-intercepts	Critical Points	Concavity
$f(x) = x^2 + 5x - 6$	$(0, -6)$	$(1, 0)$, $(-6, 0)$	$(-2.5, -12.25)$ (min)	Upward (everywhere)
$f(x) = (1 - 2x)^3$	$(0, 1)$	$(\frac{1}{2}, 0)$	$(\frac{1}{2}, 0)$ (inflection)	Changes at $x = \frac{1}{2}$

Key Concepts in Curve Sketching

Intercepts

• X-intercept: The point(s) where the graph crosses the x-axis ($f(x) = 0$).

• Y-intercept: The point where the graph crosses the y-axis ($x = 0$).

Critical Points

• Critical Point: Where $f'(x) = 0$ or $f'(x)$ is undefined. May be a local maximum, minimum, or inflection point.

• Second Derivative Test: If $f''(x) > 0$ at a critical point, it is a local minimum; if $f''(x) < 0$, it is a local maximum; if $f''(x) = 0$, further analysis is needed.

Concavity and Inflection Points

• Concavity: Determined by the sign of $f''(x)$. Positive means concave up; negative means concave down.

• Inflection Point: Where the concavity changes ($f''(x) = 0$ and sign of $f''(x)$ changes).

Applications in Business Calculus

• Understanding the shape of cost, revenue, and profit functions.

• Identifying optimal points for maximizing or minimizing business objectives.

Additional info: Asymptotes are not present in these polynomial examples, but are important for rational and exponential functions in business calculus.