Megan Langford
We
know how linear functions operate.
The slope indicates the steepness of the line, and the additive term
indicates the y intercept. But
what happens when we combine two linear functions via some operation? LetŐs explore using the following two
functions.
We
know that the pink function increases by two units vertically for every one unit
increase horizontally, and we can see this behavior in the graph. Additionally, we know that the y
intercept is 1, since this is our additive term. The same is true for the red function, since the line
appears to increase by 3 vertical units for every 1 unit horizontal
increase. In fact, the y intercept
does appear to be -2, which is our additive term in this case.
Adding the Functions
So
what happens if we add these two functions? Mathematically, we have . So the
graph should increase by 5 vertical units for every horizontal unit increase,
and it should cross the y axis at y=-1.
Indeed,
our predictions have proven to be true.
Multiplying the Functions
So
now, what would we get if we were to multiply the two functions? Mathematically, we would arrive at . This is
a quadratic equation. LetŐs take a
look.
Indeed
because this is a quadratic equation, our resulting graph is in the shape of a
parabola. Because our leading
coefficient is a rather large value of 6, the parabola takes a narrow
shape. Additionally, because this
value is positive, the parabola is oriented up rather than down. The -2 final term also indicates the y
value at the y intercept.
Further
research and analysis would tell us that the x value of the vertex is located
at . By
plugging this value in to the equation, we can see that this means the y value
is , so our vertex lies at .
Dividing the Functions
So
then what would be the effect of dividing one function by the other? Mathematically, we would get . How
about the graph? Well, we know
that the function would be undefined at , since this would result in a 0 value for the
denominator. What other behavior
could we observe?
We
can see there does appear to be a vertical asymptote where .
Additionally, as the absolute value of x gets larger and larger, the y
value gets closer and closer to 0, since the numerator will be smaller than the
denominator. This is why the y
value approaches 0 on both the right and left tails of the graph as x gets
further away from the center of the graph.
Function Composition
So
what happens when we make one of the functions a composition of the other? Well, we end up with either , which simplifies to , or , which simplifies to . Because
these functions share the same slope, we know that they will be parallel to
each other. Also, the first
function will cross over the y axis at y=-3, and the second function will cross
over the y axis at y=1. LetŐs
check to see that this is the case.
Indeed
our assumptions have proven to be accurate, and we now can see how much
different the operation can make on the outcome of a graph.