Determine Function Behavior (Curve Sketching)
1. find critical points (y’=0 or y’ DNE)2. test the regions to see if function incr or decr3. find possible inflection points (y”=0 or y” DNE)4. test for concavity5. sketch the curve (with other helpful pts)
When sketching a curve, it’s important to consider the behavior of the function in relation to its inputs. Here are some things to look for:
1. Domain and Range: The domain is the set of all valid input values for the function, while the range is the set of all possible output values. These can help determine the overall shape of the curve, as well as any points or areas where the function is undefined.
2. Symmetry and Asymptotes: If the function is symmetric with respect to the y-axis, x-axis, or origin, this can help simplify the sketching process. Additionally, any vertical or horizontal asymptotes can provide information about where the function approaches infinity or negative infinity.
3. Critical Points: The critical points of a function occur where the derivative is equal to zero, or where it does not exist. These can indicate where the function changes direction (i.e. from increasing to decreasing, or vice versa), as well as where any local maxima or minima occur.
4. Concavity and Inflection Points: The second derivative of a function can tell us whether the curve is concave up or down, which can help determine the shape of any humps or dips in the curve. Inflection points occur where the concavity of the curve changes.
By taking these factors into account, you can get a better understanding of how the function behaves and create a more accurate curve sketch.
More Answers:
Exploring The Line Of Symmetry: The Graph Of A Function And Its Inverse CompositionWhy The Composition Of Functions Is Not Commutative: The Basics Of Function Composition
Ensuring Well-Defined Compositions: Domain Restrictions In Function Composition – A Math Perspective