Curve Sketching is the process of using the first and second derivative and information gathered from the original equation to graph a function.

Steps

1. Determi ne the Domain and Range
2. Test for x, y, and origin symmetry
3. Find any x- and y- intercepts
4. Determine any vertical and horizontal asymptotes
5. Find the first derivative and use the first derivative test to test for intervals of increase & decrease and relative extrema
6. Find the second derivative and use the concavity test to test for points of inflection and changes in concavity
7. Use the information gathered and points from the original equation to draw the curve

Examples

Noncalculus

With Calculus

Graphing Using the First Derivative

Derivative Graph

Using the Derivative Graph find:
1. The critical numbers and the relative extrema
2. Intervals of increase and decrease
3. Inflection points and where the graph is concave up and concave down

From the Derivative Graph we can see:
1. There are critical numbers at x=-2, 1. At x=-2 the derivative changes from - to + so it is a relative min. At x=1 the derivative graph changes from + to - so it is a relative max.
2. The intervals of increase are where the derivative is positive. That interval is (-2,1). The interval of decrease are where the derivative is negative. That interval is (-3,-2) and (1,2.5)
3. The inflection points are where the derivative changes from increasing to decreasing or decreasing to increasing. The intervals of increase to decrease are at x=-1,2. The graph is concave up on the interval (-3,-1) and (2,3) because the derivative is increasing. The graph is concave down on the interval (-1,2) because the derivative is decreasing.

## Table of Contents

Curve Sketching is the process of using the first and second derivative and information gathered from the original equation to graph a function.Definition

1. Determi ne the Domain and RangeSteps2. Test for x, y, and origin symmetry

3. Find any x- and y- intercepts

4. Determine any vertical and horizontal asymptotes

5. Find the first derivative and use the first derivative test to test for intervals of increase & decrease and relative extrema

6. Find the second derivative and use the concavity test to test for points of inflection and changes in concavity

7. Use the information gathered and points from the original equation to draw the curve

ExamplesNoncalculusWith CalculusGraphing Using the First Derivative

Derivative Graph

Using the Derivative Graph find:

1. The critical numbers and the relative extrema

2. Intervals of increase and decrease

3. Inflection points and where the graph is concave up and concave down

From the Derivative Graph we can see:

1. There are critical numbers at x=-2, 1. At x=-2 the derivative changes from - to + so it is a relative min. At x=1 the derivative graph changes from + to - so it is a relative max.

2. The intervals of increase are where the derivative is positive. That interval is (-2,1). The interval of decrease are where the derivative is negative. That interval is (-3,-2) and (1,2.5)

3. The inflection points are where the derivative changes from increasing to decreasing or decreasing to increasing. The intervals of increase to decrease are at x=-1,2. The graph is concave up on the interval (-3,-1) and (2,3) because the derivative is increasing. The graph is concave down on the interval (-1,2) because the derivative is decreasing.

Original Graph

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