Curve Sketching

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Back in the day, curve sketching by hand was an important part of precalculus. But with the advent of the graphing calculator, sketching curves by hand isn’t usually necessary any more. Graphing calculators are allowed on most calculus exams (even AP Calculus), so you can graph your function on the TI-89 to get an idea of the overall shape. Then it’s just a matter of labeling the curve.

You may be asked to specifically include:

  1. X-intercepts,
  2. Y-intercepts,
  3. Horizontal asymptotes,
  4. Vertical asymptotes,
  5. Critical Points (maximums and minimums),
  6. Points of inflection,
  7. Endpoints.

See also: Using stationary points for graph sketching.

Read on for detailed information about how to find each of these important graph parts.

1. Curve Sketching: Finding X-intercepts

Cubic Function With Three Real Roots
A cubic function with three roots (places where it crosses the x-axis).

X-intercepts are where the function crosses the x-axis. These points are called roots or zeros. There are many ways to find roots, including on the TI-89 and using the rational number theorem. See the main roots/zeros article for more details on how to solve for a function’s roots.

You can also find roots for curve sketching with the quadratic formula. This requires you to have some strong algebra skills (including the ability to recognize when you can use the quadratic formula, and when you cannot).

curve sketching
The quadratic formula.

Example question: Find the zeros of this function algebraically: f(x) = x2 – 10x + 16

  1. Set f(x) to zero: 0 = x2 – 10x + 16,
  2. Identify a, b, and c. A quadratic function, like this one, has the form ax2 + bx + c = 0. So, a = 1, b = -10, c = 16.
  3. Plug a, b, and c values into the quadratic formula: quadratic formula for curve sketching
  4. Solve, using algebra, to get two roots: 8 and 2.

If you’re algebra is a little rusty, check out Symbolab’s calculator, which shows the steps for this solution.

2. Finding Y-intercepts

y intercept examples
Examples of y-intercepts. The graph on the right, with two y-intercepts, isn’t a function.

The next important topic in curve sketching: you’ll want to find where your graph crosses the Y-axis. To find the y-intercept, set x to zero and solve.

For example: y = x2 + 5x
Set x to zero and solve: y = 02 + 5(0) = 0
This function’s y-intercept is at x = 0.

For a quadratic equation, the y-intercept is the point “c”:
y = ax2 + bx + c

For example, y = 10x2 + 5x + 9, the y-intercept is 9.

In calculus, you won’t see more than one y intercept, because it creates a major issue called “one to many.”

3. Horizontal asymptotes

See: Horizontal asymptotes
How to Find Asymptotes

4. Vertical asymptotes

How to Find Asymptotes

5. Critical Points (maximums and minimums)

How to Find Critical Points.
Finding minima and maxima with the second derivative test.

6. Points of inflection

See: Points of inflection

7. Endpoints

See: Endpoints

Curve Sketching: References

Khan, D. Cracking the AP Calculus AB & BC Exams. Random House Information Group. 2009.

Stephanie Glen. "Curve Sketching" From Elementary Statistics for the rest of us!

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