Hyperbolic function

LO 2 : Functions and Algebra The learner is able to investigate, analyse, describe and represent a wide range of functions and solve related problems

1. Investigate the effect of a =+/-1 on the graph : 2. Investigate the effect of a on the graph: 3. Investigate the effect of q on the graph of: 4. Use reciprocal equations to solve real – life problems.

The general form of the hyperbola (rectangular function) The general form is: The standard form is:

a is a constant and can be positive or negative, but can never be equal to zero. The x and y –axes are asymptotes. Asymptotes are lines that the graph tends towards (comes extremely close to it) but never touches. There are no x or y-intercepts. The graph has two branches, either in the first and third quadrant or in the second and fourth quadrant. As the x - values increases y - values decreases if a>0 and it means the variables are indirectly or inversely proportional. The most accurate way of sketching this graph is to set up a table. The general form of the hyperbola

Investigate the effect of a =+/-1 on the graph : If a=1 Branches are in the quadrants where both x and y have positive signs or where both are negative and are therefore positive quadrants Both graphs are symmetrical to the lines y = x and y = -x y x

Investigate the effect of a =+/-1 on the graph : As the x – values increase, the y – values decrease at the same time. Domain: Range: y x

If a = -1, the graph lies in the second and fourth quadrant: Both graphs are symmetrical to the lines y = x and y = -x and lies in quadrants where the x and y values have opposite signs and are therefore negative quadrants.

Test Your Knowledge 1. Sketch the graph of y =, if x>0

Solutions y x

Investigate the effect of a on the graph: If a is small, but a>0, the hyperbola gets closer to the origin and moves further away if a becomes bigger. y x

If a<0 and a becomes bigger, the graph also moves closer to the origin ( in quads II and IV): Remember: -6 < -4 < -1 x

Test Your Knowledge 1. Sketch the graphs of: and

x y Solution y x x y

Investigate the effect of q on the graph: The vertical asymptote is always the y – axis, because: The value of q determines the horizontal asymptote. If the value of q changes, the graph moves up or down and it leads to a vertical translation of the graph. The salient point is the point on the graph where the line meets if a>0.

Investigate the effect of q on the graph: The salient points on the graph f on the next slide is (1;-1) and (-1;1) If 2 is added to the equation of f, then for the new graph: g, the salient point moves 2 units upwards to (1;1) and (-1;3).

The graphs of f and g are sketched below. Note how the axis of symmetry changes from the lines: x

Determining the Equation of a Hyperbolic Function Use the equation: First substitute the asymptote into q and then substitute into x and y and calculate a.

Test your knowledge

Use reciprocal equations to solve real – life problems. John has to wrap gifts for a year-end function of a large company. If he works alone it will take him 10 hours to finish the job. He invited a few of his friends to assist him with this task. The following table shows a clearer picture of the information, if we suppose they all work at the same rate: Number of PeopleTime (hours) to complete task hrs 20 min 42 hrs 30 min 52 hours John used the formula:

The graph is a hyperbolic function and occurs only in the first quadrant. Why? y x

Test Your Knowledge 1. On the same system of axis, sketch the graph of: and, clearly showing all asymptotes and lines of symmetry.

Solution y x

Test your knowledge