Sketching Curves
Sketching Curves
Understanding Curves vs. Straight Lines

The difference between a curve and a straight line is that a straight line has a constant slope whereas a curve does not.

Any point on a curve can have a tangent  a straight line that “just touches” the curve at that point.

The slope of the tangent at a point gives the derivative of the function at that point.
Sketching Curves

Sketching curves, unlike straight lines, usually involves dealing with curves that are described by more complex function.

For a beginner, graphing a curve might include plotting a large number of points and then sketching a line that approximates those points.

Essential curves to be able to sketch are parabolas, cubic functions, circle equations, and other higher order polynomials.
Using a Table of Values

A table of values can be used to sketch a curve by inputting different xvalues into a function to find the corresponding yvalues.

Plotting these pairs of values on a graph and then joining them to form a smooth curve provides a visual representation of the function.

A table of values helps when sketching curves which don’t have an easy algebraic equation, or for checking the accuracy of your graph.
Calculating Turning Points

A curve’s turning point is a point at which the derivative of the function equals zero.

They are also known as local maxima or local minima because they represent the highest or lowest point on the graph, respectively, in their vicinity.

Calculating turning points can help you to accurately sketch the shape and important points of a curve.
Using Symmetry

Some curves, such as parabolas and cubic functions, demonstrate symmetry.

Parabolas are symmetric about a vertical line, known as the line of symmetry, that passes through the vertex of the parabola.

Symmetry can be utilised to simplify the process of sketching curves, by drawing half of the curve first, and then reflecting it to complete the graph.
Using Asymptotes

Asymptotes are lines that a curve approaches but never crosses, providing valuable information about the behaviour of the curve at extreme values.

Recognizing and sketching asymptotes can provide a backbone for your graph and help sketch complicated functions more accurately.

Horizontal asymptotes represent the function’s behaviour far to the left or the right on the xaxis, whereas vertical asymptotes signify a division by zero in a function.