Sketching curves

Basic sketching skills: Understand how to create a simple sketch of a function’s graph. Pay attention to the curve, the x and y intercepts, turning points and points of inflection.
Symmetry: Learn to identify symmetry in a curve. If f(-x) = f(x), the graph is symmetric about the y-axis. If f(-x) = -f(x), it is symmetric about the origin.
Asymptotes: Familiarize yourself with the concept of vertical and horizontal asymptotes and how they impact the shape of a graph. Vertical asymptotes are y values that the function never reaches, whereas horizontal ones are x values.
Increasing and decreasing functions: An increasing function is one where y increases as x increases. A decreasing function is one where y decreases as x increases.
Turning points: These are points where the gradient of a function is zero. If the second derivative at this point is positive then it’s a minimum turning point, negative and it’s a maximum.
Inflection points: An inflection point is a point where the curvature of a function changes sign. These points often indicate important characteristics of the function.
Regions of interest: Identify regions of a graph that have particular properties – for instance, where the function is positive or negative, or where it is increasing or decreasing.
Interpretation: Be prepared to interpret what the graph can tell you about a real world situation. For instance: When does a model predicted by the function reach its maximum value? How does a change in variable affect the outcome?
Higher degree polynomials: Understand the shape of higher degree polynomials, which may have multiple turning points and inflection points. As a general rule, they look like waves, with amplitude increasing as you move away from the origin.
Trigonometric functions: Familiarize yourself with the graph shapes of the sine, cosine and tangent functions, their heights, periods, frequencies and phase shifts.
Exponential and logarithmic functions: Understand how the graphs of these functions look and behave. Typically, they grow or decay very quickly.
Rational functions: Know that these are functions which can be expressed as a ratio of two polynomials. Understand how to identify any vertical asymptotes and horizontal asymptotes these graphs may have, as well as how to locate any holes.
Transformed graphs: Comprehend how transformations such as shifts, stretches and reflections affect a graph. These transformations can change the location, size and orientation of a graph in a predictable way.
Use software when needed: Some complex functions might be difficult to sketch by hand. Know that graphing tools and software can sometimes provide the most accurate graphs.
Practice: Practice with a variety of different types and complexities of curves.

Remember: It’s not always about accuracy, but about capturing key characteristics of a function’s graph.