Now explore the graph of h(x), where h(x) = f(x) +
g(x). Notice how this graph relates to the original
Notice the slope of this new function appears to be very similar to
that of g(x). Algebraically, this function can be shown as h(x) =
-(14/5)x -2. The slope is -14/5 and the y - intercept
Let's now explore the graph of h'(x), where h'(x)
Here we can see that the graph of two linear functions multiplied
together is in this case a parabola.
Let's now explore the graph of h''(x) where h''(x) = f(x)/g(x).
Here we notice that the graph is a hyperbola.
Lastly, look at the graph of h'''(x), where h'''(x) = f(g(x)).
The compositions of these two functions, f of g of x, is a linear
Now, let f(x) and g(x)
different functions and explore these graphs again.
Let f(x) = 5x + 2,
and let g(x) = x/2 + 1.
Below are the graphs of these two
equations. f(x) is the purple graph and g(x) is the red graph.
h(x) = f(x) +
g(x) is shown by the blue function. Again, notice how the process
of addition causes the original equation with the largest absolute
value to dominate.
= f(x).g(x) Again, when we multiply two linear equations
together, we get a parabola.
h''(x) = f(x)/g(x) Dividing one linear equation
by another yields a hyperbola.
h'''(x) = f(g(x)) The composition of two linear functions, f of g
of x, creates a new function, also linear.
Summary of Findings
When two linear functions are added together, the result is a
third linear function. The slopes and y-intercepts of the functions
follow the basic properties of addition when added together,
When two linear functions are multiplied together, the resulting
will be a parabola, provided that the slope of neither function is
0. In that case, the resulting function will either be a linear
function (if one function has zero slope) or a constant function (if
functions have zero slopes.) Likewise, the resulting function
one linear function is divided by another will be a hyperbola, provided
neither function has a zero slope. If a linear function with a
zero slope is divided by a function with a zero slope, the resulting
function will also be linear.
The composition of two linear functions always yields another linear
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