The Easiest Way to Graph Any Function Using Derivatives

Lesson 10 of the Calculus 1 series: how to sketch a graph by combining what f(x), f′(x), and f″(x) each tell you. Learn how to sketch graphs using derivatives in this step-by-step calculus lesson. We use the original function, the first derivative, and the second derivative to extract the information needed to understand and graph a function. In this lesson, we cover: • Finding the x-intercepts and y-intercept • Determining where a function is above or below the x-axis • Using the first derivative to find local maximum and minimum points • Identifying where a function is increasing or decreasing • Using the second derivative to determine concavity • Finding inflection points • Finding vertical, horizontal, and slant asymptotes of rational functions • Using derivatives to sketch rational functions • Graphing a cubic function using its first and second derivatives • Understanding how the signs of the first and second derivatives determine the shape of a graph The lesson includes a detailed rational-function example, including a vertical asymptote, horizontal asymptote, intercepts, increasing/decreasing behavior, and concavity. We also work through the cubic function y = x³ − 9x² + 24x − 10 and use its first derivative to locate critical points and its second derivative to identify concavity and the inflection point. What You Will Learn By the end of the lesson, you will see how f(x), f′(x), and f″(x) work together to reveal the behavior and shape of a function—even when finding the exact x-intercepts is difficult. This is an important foundation for Calculus 1 graphing, curve sketching, derivatives, extrema, concavity, and optimization. Subscribe to Math and Engineering Made Easy for more step-by-step lessons in calculus, mathematics, engineering, and problem solving. #Calculus #CalcOne #CurveSketching #GraphingFunctions #Derivatives #InflectionPoint #Concavity #MathTutor #LearnCalculus #APCalculus #APCalcAB #APCalcBC #MathAndEngineeringMadeEasy #CalculusForBeginners #STEM