How to Graph Variables Directly and Inversely

An inverse function essentially undoes the work of the original function. If the original function doubles all the values of x, the inverse divides them by two. One common use for inverse functions is solving equations, because isolating a variable is like undoing what is being done to it. To calculate an inverse function, take the original equation, swap every x for a y and solve for y. Technically, no function can have more than one value of y for any given value of x, so some functions don't have inverses.

Instructions

    • 1

      Make a list with two columns for x- and y-coordinates. In the x column, fill in values of x you want to show on your graph. In the y column, use the given function to calculate the values of y corresponding to each value of x. For example, if your function is y = 2^x, your table might be:

      x, y

      -10, 0.001

      -5, 0.031

      -3, 0.125

      -2, 0.25

      -1, 0.5

      0, 1

      1, 2

      2, 4

      3, 8

      4, 16

    • 2

      Draw the x- and y-coordinate axes and label the axes with values so that most of your points will fit on the graph. In the given example, you might choose values ranging from -10 to 10 for both the x- and y-axes since the point (4, 16) is the only one that will not fit. Calculating (4, 16) is still useful because it gives you a better idea of the graph's direction after (3, 8).

    • 3

      Plot the x- and y-coordinates on the graph and connect them with a smooth curve. Extend the curve to the edge of the graph. If you are unsure about how the graph should look at a particular point, calculate more coordinates as needed.

    • 4

      Plot the inverse function by swapping every x for a y. In the given example, the inverse function includes the points (0.001, -10), (0.008, -7), (0.031, -5), etc. Algebraically, it is y = log2(x), where log2 signifies the base-2 logarithm of x. Connect the points with a smooth curve as you did with the original function. The inverse function should look like a mirror image of the original function across the line y = x.

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