This lesson is meant to summarize all the major concepts covered from this unit. The original function and its respective derivatives are used to find different pieces of information about the function. We can combine all of this information to help draw an accurate sketch of the original function.
From \(f(x)\):
From \(f'(x)\):
From \(f''(x)\):
In order to identify the \(x\)-intercept, we need to set the function (or numerator if it's rational) equal to \(0\). We can then solve for \(x\). If we can't cleanly factor the function, then we can use either the Quadratic Equation or use Synthetic Division to find the roots.
In order to identify the \(y\)-intercept, we can substitute \(x = 0\) and solve for \(y\).
Asymptotes only exist for Rational Functions.
In order to determine the Vertical Asymptote, we can set the denominator equal to \(0\). We can then solve for \(x\). The \(x\)-intercepts cut the \(x\)-axis based on the exponent (ie cut, bounce, squiggle).
Horizontal Asymptotes only exist if the degree of the numerator and denominator are the exact same. If so, then find the ratio of their leading coefficients.
Oblique Asymptotes only exist if the numerator is exactly 1 degree higher than the numerator. If so, then use long division to find its equation.
At the end, find all y-values of all the Special Points outlined above in order to plot them accurately.
In order to determine the Critical Points, we can find the first derivative of the original function, \(f'(x)\). We can then factor this expression and set it equal to 0 to find its roots. If we can't factor this function cleanly, then we can use either the Quadratic Equation or Synthetic Division, similar to finding the \(x\)-intercepts.
In order to identify the Intervals of Change, we can use the First Derivatives Test. From this, we can determine which intervals the original function is increasing and which its decreasing. If the function is increasing (+) then decreasing (-), this indicates a Local Maximum. If the function increases (+) then decreases (-), then that represents a Local Minimum.
In order to identify the Concavity of a function, we can find the Second Derivative of the original function, \(f''(x)\). We can then factor this expression and set it equal to 0 to find its roots. If we can't factor this function cleanly, then we can use either the Quadratic Equation or Synthetic Division, similar to finding the \(x\)-intercepts or Critical Points.
In order to determine the actual Point(s) of Inflection, we can conduct the Second Derivatives Test. From this, we can determine the respective concavities of each interval. If the concavity changes from positive (+) to negative (-) in between intervals, this represents a Point of Inflection.
Enter in a coefficient for quadratic monomial or click on the New Question button for a random value. Then, use the First Principles formula to solve for the derivative.