This week in Pre-calculus 11 we learned more about absolute values and reciprocal functions. One thing that I learned that really stuck with me was when we learned about reciprocal functions and how to graph them.

Things to know:

• A positive reciprocates to a positive (+ –> +)
• A negative reciprocates to a negative (- –> -)
• When a big number is reciprocated it turns into a small number and vise versa.
• When both 1 and -1 are reciprocated they stay the same (called invariant points)
• When 0 is reciprocated it becomes undefined (you can’t have a 0 on the bottom of a fraction)
• Asymptote – horizontal/vertical line that separates the graph. The graph will approach both lines but never touch them. (the vertical asymptote is located halfway in-between the two invariant points/Horizontal will always be y=0).
• Hyperbola – two curves similar to each other, indicated by the similar points. Opposite of each other. Curved lines on the graph.
• The two hyperbolas will never touch due to the vertical asymptote

Example:

• First, we graph the linear function as we normally would if it wasn’t reciprocated. To do this we would write down everything we know about the original function and plot it on the graph. Since the original function is 2x+6, we know that the function will be positive and have a slope of 2, we also know that the y-intercept will be 6.
• Next, you circle your invariant points which will be when y is at both 1 (_,1) and at -1 (_,-1). To do this go to where 1 is on the y-axis and then slide your finger vertically along the graph until you touch the linear function. You would do the same thing for -1.
• Next you draw in the asymptotes. Since the horizontal asymptote will always be y=0, we draw a horizontal broken line along that part of the graph. Also, since the vertical line is half way in between the two invariant points, we know it will be when x= -3. Another way to find what the vertical asymptote is without looking at a graph is by making the original linear function equal to zero and solving for x.
• After drawing in the asymptotes, you can now draw the hyperbolas. To do this you would take of the points along the linear function, reciprocate it and plot its reciprocal. You would then draw the hyperbola through the invariant and the new reciprocated points.

Note: when writing the domain or range x and y can be any real number except they can’t be equal to the corresponding asymptote.