# 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 x-values into a function to find the corresponding y-values.

• 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 x-axis, whereas vertical asymptotes signify a division by zero in a function.