Curve sketching

Understanding the Goal of Curve Sketching

Curve sketching is about drawing a reasonably accurate graph of a function using theoretical information (derivatives, limits, etc.), not a point-by-point table or a calculator.

In this chapter, the focus is on a systematic procedure: what information to extract from a function, how to organize it, and how each piece affects the shape of the graph.

We will assume you already know:

• How to compute first and second derivatives.
• The ideas of increasing/decreasing, local maxima/minima, and concavity from the general “Applications of Derivatives” chapter.
• Basic limit ideas (including what happens near asymptotes) from earlier calculus chapters.

Here we put these ideas together into a step-by-step sketching method.

A General Strategy for Curve Sketching

Given a function $f(x)$, a standard curve-sketching strategy is:

1. Determine the domain.
2. Find intercepts (where the graph meets the axes).
3. Locate symmetry (even, odd, periodic, etc., if applicable).
4. Look for asymptotic behavior (vertical, horizontal, oblique).
5. Use the first derivative to find:
• Critical points.
• Intervals of increase/decrease.
• Local maxima/minima.
6. Use the second derivative to find:
• Intervals of concavity (up/down).
• Inflection points.
7. Combine all this information into a coherent sketch.

Not every function needs all steps (some have no asymptotes, some have no symmetry, etc.), but this checklist keeps you from missing something important.

We now examine each part in the context of sketching.

Step 1: Domain and “Where the Graph Exists”

The domain tells you the $x$-values for which $f(x)$ is defined. For curve sketching, this is the starting constraint: the graph simply does not exist outside the domain.

Typical restrictions you might encounter:

• Division by zero:
• If $f(x) = \dfrac{1}{x-2}$, then $x \ne 2$.
• Even roots of negative numbers (when considering real-valued functions):
• If $f(x) = \sqrt{4-x}$, then $4-x \ge 0$, so $x \le 4$.
• Logarithms of nonpositive numbers:
• If $f(x) = \ln(x-1)$, then $x-1 > 0$, so $x > 1$.

In practice for curve sketching:

• Write the domain as an interval or union of intervals.
• Mark excluded points (like $x = 2$) or endpoints (like $x = 4$) clearly on your mental picture or scratch graph.

The domain often hints where vertical asymptotes or endpoints of the graph might be.

Step 2: Intercepts

Intercepts anchor the graph.

• $y$-intercept: where the graph crosses the $y$-axis ($x = 0$).
• Compute $f(0)$, if $0$ is in the domain.
• $x$-intercepts: where the graph crosses the $x$-axis ($y = 0$).
• Solve $f(x) = 0$ (if possible).

Examples:

• If $f(x) = x^3 - 3x$, then $f(0) = 0$ is both the $x$- and $y$-intercept.
• If $f(x) = \dfrac{x+1}{x-2}$, the $y$-intercept is $f(0) = -\tfrac12$, and the $x$-intercept solves $x+1=0$, so $x=-1$.

Even if exact intercepts are hard to find, sometimes approximate values are enough to place them qualitatively on the graph.

Step 3: Symmetry Considerations

Symmetry can reduce the work:

• Even function: $f(-x) = f(x)$ for all $x$ in the domain.
• Graph is symmetric about the $y$-axis.
• Example: $f(x) = x^2$.
• Odd function: $f(-x) = -f(x)$.
• Graph is symmetric about the origin (a $180^\circ$ rotation about the origin).
• Example: $f(x) = x^3$.
• Periodic function: repeats over intervals.
• Example: $f(x) = \sin x$ has period $2\pi$.

To use this in sketching:

• If $f$ is even, you can analyze $x \ge 0$ and reflect across the $y$-axis.
• If $f$ is odd, you can analyze $x \ge 0$ and rotate the graph around the origin.
• If $f$ is periodic, sketch one period carefully and then repeat.

For many arbitrary functions, there is no symmetry; in that case, you simply proceed without it.

Step 4: Asymptotic Behavior

Asymptotes guide the behavior of the graph “at infinity” or near points where the function blows up.

Vertical Asymptotes

A vertical line $x = a$ is a vertical asymptote if the function grows unbounded as $x$ approaches $a$ from one or both sides, for example:
$$
\lim_{x \to a^-} f(x) = \pm \infty \quad \text{or} \quad \lim_{x \to a^+} f(x) = \pm \infty.
$$

Typical sources:

• Denominator $= 0$ while numerator is nonzero (for rational functions).
• Logarithmic functions: $\ln x$ has a vertical asymptote at $x=0^+$.

In sketching, you:

• Draw a dashed vertical line at $x = a$.
• Indicate whether $f(x)$ goes to $+\infty$ or $-\infty$ on each side.

Horizontal Asymptotes

A horizontal line $y = L$ is a horizontal asymptote if
$$
\lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L.
$$

This describes the long-term behavior as $x$ becomes large (positive or negative).

For many rational functions, you can discover horizontal asymptotes by comparing degrees of numerator and denominator (though the details of that belong more to function analysis and rational functions).

In sketching, draw a dashed horizontal line $y = L$ and understand that the graph approaches it but may cross it.

Oblique (Slant) and Other Asymptotes

Sometimes the graph approaches a non-horizontal line like $y = mx + b$ as $x \to \pm\infty$; this is an oblique asymptote. For example, some rational functions where the degree of the numerator is one higher than the denominator.

In sketching, you can:

• Draw the line $y = mx + b$ as a dashed guide.
• Show the curve approaching it at large $|x|$.

The main point: asymptotes shape the overall “skeleton” of the graph, especially far from the origin or near undefined points.

Step 5: First Derivative – Increasing/Decreasing and Extrema

The first derivative provides information about the slope and therefore about increasing/decreasing behavior and local maxima/minima.

Critical Points

A critical point (in one variable) is an $x$ in the domain of $f$ where:

• $f'(x) = 0$, or
• $f'(x)$ does not exist (but $f$ itself is defined at $x$).

Critical points are where local maxima or minima might occur. (They also include flat spots that are not maxima or minima.)

To use them in curve sketching:

1. Compute $f'(x)$.
2. Solve $f'(x) = 0$.
3. Identify where $f'(x)$ is undefined but $f(x)$ is defined.
4. Collect all such $x$-values as potential “special points”.

Increasing and Decreasing Intervals

To understand the behavior between critical points:

• Make a sign chart for $f'(x)$.
• Pick test points between consecutive critical points and determine the sign of $f'(x)$ in each interval.

If $f'(x) > 0$ on an interval, $f$ is increasing there.
If $f'(x) < 0$ on an interval, $f$ is decreasing there.

From these sign changes, you can identify local maxima and minima.

• If $f'(x)$ goes from $+$ to $-$ at a critical point $c$, then $f$ has a local maximum at $x = c$.
• If $f'(x)$ goes from $-$ to $+$ at $c$, then $f$ has a local minimum at $x = c$.
• If $f'(x)$ does not change sign, the critical point is not a local extremum (it could be a flat inflection, for instance).

In the sketch, mark these points $(c, f(c))$ clearly and note whether each is a local max or min.

Step 6: Second Derivative – Concavity and Inflection Points

The second derivative provides information about concavity—whether the graph curves upward or downward.

Concavity

• If $f''(x) > 0$ on an interval, the graph is concave up there (“cup” shape; slopes are increasing).
• If $f''(x) < 0$ on an interval, the graph is concave down there (“cap” shape; slopes are decreasing).

To find concavity intervals:

1. Compute $f''(x)$.
2. Solve $f''(x) = 0$ and find where $f''(x)$ does not exist (but $f$ exists).
3. Use a sign chart for $f''(x)$ across those points.

Inflection Points

An inflection point is a point on the graph where concavity changes (from up to down, or down to up).

• Candidate $x$-values are where $f''(x) = 0$ or $f''(x)$ is undefined.
• To confirm an inflection point at $x = c$, check that the sign of $f''(x)$ changes around $c$ (and that $f$ is defined at $c$).

In the sketch:

• Plot inflection points $(c, f(c))$.
• Show the change from one concavity to the other.

Step 7: Putting It All Together – Building the Sketch

Once you have:

• Domain.
• Intercepts.
• Symmetry (if any).
• Asymptotes.
• Intervals of increase/decrease, with local maxima/minima.
• Intervals of concavity and inflection points.

You can construct a reasonably accurate sketch as follows:

1. Draw the axes and asymptotes:
• Mark the $x$- and $y$-axes.
• Draw any vertical, horizontal, or oblique asymptotes with dashed lines.
2. Plot key points:
• Intercepts.
• Local maxima and minima.
• Inflection points.
• Any notable endpoints of the domain.
3. Indicate intervals:
• On the $x$-axis (or mentally), note where the function is increasing/decreasing and where it is concave up/down.
4. Connect smoothly:
• Draw a smooth curve through the key points, obeying:
• Increasing/decreasing behavior.
• Concavity changes.
• Asymptotic behavior (approach asymptotes appropriately).
• Make sure the curve does not suddenly break any of these rules.

The goal is not perfection but a graph that respects all the known qualitative behaviors.

Worked Example: Sketching a Polynomial

Consider $f(x) = x^3 - 3x$.

We outline the curve-sketching steps; details like basic derivative computation are assumed.

1. Domain: All real numbers (polynomials are defined everywhere).
2. Intercepts:
• $y$-intercept: $f(0) = 0$.
• $x$-intercepts: Solve $x^3 - 3x = 0$:
  $$
  x(x^2 - 3) = 0 \quad \Rightarrow \quad x = 0, \; x = \pm \sqrt{3}.
  $$
  So intercepts at $(-\sqrt{3}, 0)$, $(0,0)$, $(\sqrt{3}, 0)$.
3. Symmetry:
• $f(-x) = (-x)^3 - 3(-x) = -x^3 + 3x = -(x^3 - 3x) = -f(x)$.
• So $f$ is odd; graph is symmetric about the origin.
4. Asymptotes:
• Being a polynomial, there are no vertical or horizontal asymptotes, and no oblique asymptotes.
5. First derivative:
• $f'(x) = 3x^2 - 3 = 3(x^2 - 1)$.
• Critical points: $f'(x) = 0 \Rightarrow x^2 - 1 = 0 \Rightarrow x = \pm 1$.
• Sign chart for $f'(x)$:
• For $x < -1$: $x^2 > 1 \Rightarrow f'(x) > 0$ (increasing).
• For $-1 < x < 1$: $x^2 < 1 \Rightarrow f'(x) < 0$ (decreasing).
• For $x > 1$: $x^2 > 1 \Rightarrow f'(x) > 0$ (increasing).
• Local extrema:
• At $x = -1$, derivative changes $+$ to $-$ ⇒ local maximum.
• At $x = 1$, derivative changes $-$ to $+$ ⇒ local minimum.
• Corresponding $y$-values:
• $f(-1) = (-1)^3 - 3(-1) = -1 + 3 = 2$.
• $f(1) = 1^3 - 3(1) = 1 - 3 = -2$.
6. Second derivative:
• $f''(x) = 6x$.
• $f''(x) = 0$ at $x = 0$ is a candidate for inflection.
• Sign of $f''(x)$:
• For $x < 0$: $f''(x) < 0$ ⇒ concave down.
• For $x > 0$: $f''(x) > 0$ ⇒ concave up.
• $f(0) = 0$, and concavity changes at $x=0$, so $(0,0)$ is an inflection point.
7. Combine:
• Plot:
• Intercepts: $(-\sqrt{3}, 0)$, $(0,0)$, $(\sqrt{3}, 0)$.
• Local max: $(-1, 2)$.
• Local min: $(1, -2)$.
• Inflection: $(0,0)$ (already plotted).
• Mark increasing/decreasing:
• Increasing on $(-\infty, -1)$ and $(1, \infty)$.
• Decreasing on $(-1, 1)$.
• Mark concavity:
• Concave down on $(-\infty, 0)$.
• Concave up on $(0, \infty)$.
• Sketch a smooth curve consistent with these facts and symmetric about the origin.

The resulting graph has the characteristic “S” shape of a cubic with one max and one min.

Another Type: Functions with Asymptotes

Consider $g(x) = \dfrac{1}{x}$.

A brief application of the curve-sketching strategy:

1. Domain: $x \ne 0$; so $(-\infty, 0)$ and $(0, \infty)$.
2. Intercepts:
• No $y$-intercept (undefined at $x=0$).
• No $x$-intercept (equation $1/x = 0$ has no solution).
3. Symmetry:
• $g(-x) = 1/(-x) = -1/x = -g(x)$ ⇒ odd ⇒ symmetric about the origin.
4. Asymptotes:
• Vertical: $x = 0$ (as $x \to 0^\pm$, $g(x) \to \pm \infty$).
• Horizontal: $y = 0$ (as $x \to \pm\infty$, $g(x) \to 0$).
5. First derivative:
• $g'(x) = -1/x^2 < 0$ for all $x \ne 0$ ⇒ decreasing on each side of 0.
• No critical points in the domain.
6. Second derivative:
• $g''(x) = 2/x^3$:
• $g''(x) < 0$ for $x < 0$ ⇒ concave down on $(-\infty, 0)$.
• $g''(x) > 0$ for $x > 0$ ⇒ concave up on $(0, \infty)$.
• No inflection point within the domain because $x=0$ is not in the domain.
7. Combine:
• Draw vertical asymptote $x=0$ and horizontal asymptote $y=0$.
• On $(-\infty, 0)$: graph is decreasing and concave down, approaching $0$ as $x \to -\infty$ and $\pm\infty$ as $x \to 0^-$.
• On $(0, \infty)$: graph is decreasing and concave up, approaching $0$ as $x \to \infty$ and $\pm\infty$ as $x \to 0^+$.

This produces the classic hyperbola with two branches in opposite quadrants.

Common Curve-Sketching Pitfalls

When applying this method, some typical mistakes are:

• Ignoring the domain:
• Sketching the function across a point where it is not defined (e.g., drawing through a vertical asymptote).
• Misusing critical points:
• Assuming every critical point is a max or min without checking the sign changes of $f'$.
• Forgetting concavity:
• Drawing the graph bending in the wrong direction relative to the sign of $f''$.
• Misinterpreting asymptotes:
• Treating asymptotes like “hard walls” the graph can never cross. (Horizontal and oblique asymptotes can be crossed; vertical asymptotes cannot because the function is not defined at that $x$.)
• Inconsistent features:
• Drawing a local maximum where the derivative is positive on both sides, etc.

To avoid these, it helps to:

• Keep a small table of intervals with signs of $f'$ and $f''$.
• Check quickly that your final sketch agrees with that table.

Practical Tips for Curve Sketching

• Work in a consistent order (for example: domain → intercepts → asymptotes → $f'$ → $f''$).
• Label special points clearly: max, min, inflection, asymptotes.
• Do not try to be “artistically perfect”; focus on qualitative accuracy.
• For complicated algebra, locate at least:
• Approximate critical points and inflection points.
• Main asymptotes and intercepts.
• Compare with rough numeric values if needed (e.g., evaluating $f(x)$ at a few extra points) to check your picture.

Curve sketching is essentially storytelling about the function: where it lives, how it rises and falls, how it bends, and how it behaves far away. Derivatives provide the language for that story; this chapter’s strategy shows how to combine them into a coherent picture.