Curve Sketching

1. Domain

1.1. Check by setting Denominator equal to zero

1.2. Check square roots by setting them greater than or equal to zero

1.3. Otherwise, all reals

2. Intercepts

2.1. Plug in zero for the x values and solve.

2.2. Plug in zero for y values and solve.

3. Symmetry

3.1. Even- symmetric about the y-axis

3.1.1. f(-x)=f(x)

3.2. Odd- symmetric about orgin

3.2.1. f(-x)=-f(x)

3.3. Neither- if it doesn't work out to be the other two ways

3.4. Substitute in a negative x.

4. Asmyptotes

4.1. Vertical Asymptote- factor and set the denominator equal to zero, if it does those are the point where the function D.N.E.

4.2. Horizontal Asymptote- find the limits as x approaches infinity

4.3. Slant Asymptote- (y= mx+b) occur when degree of numerator is one degree bigger than the denominator, use long division

5. Intervals of Increasing/ Decreasing

5.1. Find the First Derivative

5.2. If the derivative is greater than zero, it's increasing

5.3. If the derivative is less than zero, it's decreasing

6. Sketch the curve

6.1. Use the information found, and plug in points to sketch a rough outline of the graph.

7. Local Maximums / Minimums

7.1. Find the first derivative

7.2. Factor and set equal to zero

7.3. These points are the maxes/mins

7.4. Increasing and then decreasing is a max

7.5. Decreasing and then increasing is a min

8. Concavity

8.1. Find the second derivative

8.2. If it's greater than zero, it's concave up

8.3. If it's less than zero, it's concave down

9. Points of Inflection

9.1. Find the second derivative and set equal to zero.

9.2. These are the points where the function changes between concave up and down.