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Mom's Basement University

Curve Sketching

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn how to combine everything you know about derivatives to sketch accurate graphs of functions. This is where all the derivative tools come together.

The Concept

Section titled “The Concept”

Curve sketching is a systematic process. You gather information from the function and its derivatives, then assemble it into a picture.

The checklist:

  1. Domain: where is the function defined?
  2. Intercepts: where does it cross the axes?
  3. Symmetry: is it even, odd, or neither?
  4. First derivative: where is f increasing (f’ positive) or decreasing (f’ negative)? Where are the critical points?
  5. Second derivative: where is f concave up (f” positive) or concave down (f” negative)? Where are the inflection points?
  6. Asymptotes: any vertical, horizontal, or oblique asymptotes?
  7. Sketch: put it all together.

You don’t always need every step. For polynomials, there are no asymptotes. For simple functions, symmetry might be obvious. Use what’s relevant.

Worked Example

Section titled “Worked Example”

Example 1: Full sketch of a cubic

Sketch f(x) = x³ - 3x² - 9x + 5.

Step 1: Domain is all real numbers. y-intercept: f(0) = 5.

Step 2: First derivative.

f′(x)=3x2−6x−9=3(x+1)(x−3)f'(x) = 3x^2 - 6x - 9 = 3(x + 1)(x - 3)

Critical points: x = -1 and x = 3.

Sign of f’: positive on x less than -1 (increasing), negative between -1 and 3 (decreasing), positive on x greater than 3 (increasing).

So x = -1 is a local max, x = 3 is a local min.

f(-1) = -1 - 3 + 9 + 5 = 10. f(3) = 27 - 27 - 27 + 5 = -22.

Step 3: Second derivative.

f′′(x)=6x−6f''(x) = 6x - 6

f” = 0 at x = 1. f” is negative for x less than 1 (concave down), positive for x greater than 1 (concave up). So x = 1 is an inflection point where the concavity switches.

f(1) = 1 - 3 - 9 + 5 = -6.

Putting it together: the curve rises to a local max at (-1, 10), falls through the inflection point at (1, -6), continues falling to a local min at (3, -22), then rises again.

Example 2: Concavity analysis

For f(x) = x⁴ - 4x², analyze concavity.

f′(x)=4x3−8x=4x(x2−2)f'(x) = 4x^3 - 8x = 4x(x^2 - 2) f′′(x)=12x2−8f''(x) = 12x^2 - 8

f” = 0 when 12x² = 8, so x² = 2/3, meaning x is about plus or minus 0.816.

f” is positive outside these points (concave up) and negative between them (concave down). The inflection points are at x = plus or minus sqrt(2/3).

Example 3: Quick sketch from derivative info

If you know f’ is positive on (-infinity, 2) and negative on (2, infinity), and f” is negative everywhere, then:

  • f is increasing then decreasing: local max at x = 2
  • f is concave down everywhere: the curve bends downward like an upside-down bowl
  • The graph looks like an inverted parabola shape with its peak at x = 2

Real-World Application

Section titled “Real-World Application”

Curve sketching is used whenever you need to understand the shape of a relationship:

  • Economics: sketching cost, revenue, and profit curves to find break-even points and optimal production levels
  • Physics: sketching position, velocity, and acceleration to understand motion
  • Engineering: understanding stress-strain curves to find failure points
  • Data science: understanding the shape of loss functions to know if gradient descent will converge

Quiz

Section titled “Quiz”
If f'(x) is positive on an interval, the function is:
Concave up means:
An inflection point is where:
If f' changes from positive to negative at x = c, then x = c is a:
The first step in curve sketching is usually: