Guidelines for sketching the graph of a function 4.5 Graphical Methods. Guidelines for sketching the graph of a function using first and second derivative. Rita Korsunsky

Guidelines for sketching the graph of y = f(x) There are seven guidelines to follow while sketching a graph. 1. Domain – find all x that are possible. The denominator of a function cannot equal 0. 2. Continuity – determine whether f is continuous on its domain. f is a rational function, it is continuous on its domain, but f has infinite discontinuities at –3 and 3 which are not on the domain The domain of f consists of all real numbers except –3 and 3.

3. Intercepts – find the x-and y-intercepts. The graph intersects both axes at the origin.

4. Symmetry – check to see if the function is odd or even. If the function is even, the graph will be symmetrical with respect to the y-axis. If the function is odd, the graph will be symmetrical with respect to the origin. Ex:

- - + + -3 3 f

6 a. Points of Inflection 6 b. Concavity + + + + -3 3 - + - f’’ + +

7. Asymptotes: lines graph is approaching to. + + Horizontal asymptote : y = -2

+ +

Check using graphing calculator: Time to graph! + + Check using graphing calculator:

Continuous on its domain, But it has infinite discontinuity at x=2 4.Symmetry: 2.Continuity Continuous on its domain, But it has infinite discontinuity at x=2 3.Intercepts: x-intercepts: y =0 y-intercepts: x =0

+ - 6.Points of Inflection and Concavity: 5.Critical Numbers and Extrema: + + 2 + - + +

6.Asymptotes: + +

Check using graphing calculator: Time to graph! Check using graphing calculator: + +