THE GREATEST WARM UP…
1.) Consider the table below that compares f(x) and
1
𝑓(𝑥)
. Complete any missing pieces of the
table.
x
-1
0
1
2
3
4
5
6
f(x)
1
𝑓(𝑥)
4
_______
_______
10
0
_______
positive number infinitely close to 0
_______
_______
negative number infinitely close to 0
undefined
_______
+∞
_______
_______
-∞
For what values of x is
1
𝑓(𝑥)
undefined?
For what values of x is f(x) undefined?
What are the zero(s) of f(x)?
1
What are the zero(s) of 𝑓(𝑥) ?
2.) Consider the graph of the function g(x) = x2 – 16
For what value(s) of x is the function equal to zero?
For what value(s) of x is the function infinitely close to zero but positive?
For what value(s) of x is the function infinitely close to zero but negative?
For what value(s) of x is the function infinitely large but negative?
For what value(s) of x is the function infinitely large but positive?
For each of the x value(s) identified above, explain the behavior of the reciprocal function.
Sketch a graph of the reciprocal function.
Reciprocal functions are FUN but they are limited in the fact that the numerator is always one.
This is a lot like going out to eat at the same place every night.
Consider the function h(x) = x + 6 and the function j(x) = x – 2
For h(x)….
For what value(s) of x is the function equal to zero?
For what value(s) of x is the function infinitely close to zero but positive?
For what value(s) of x is the function infinitely close to zero but negative?
For what value(s) of x is the function infinitely large but negative?
For what value(s) of x is the function infinitely large but positive?
For j(x)….
For what value(s) of x is the function equal to zero?
For what value(s) of x is the function infinitely close to zero but positive?
For what value(s) of x is the function infinitely close to zero but negative?
For what value(s) of x is the function infinitely large but negative?
For what value(s) of x is the function infinitely large but positive?
ℎ(𝑥)
Now let’s consider the ratio of h(x) and j(x)… 𝑗(𝑥)
Let’s make a table from the inputs you defined above. For each of the x value(s) identified
above, explain the behavior of the ratio function.
x
h(x)
j(x)
Behavior of
ℎ(𝑥)
𝑗(𝑥)
Other things to consider…
+Also consider the y intercept of the ratio function.
+What about the end behavior? Complete the table above for values of + ∞ and - ∞. How
does h(x) behave? j(x)? How will the ratio function behave?
MOMENT OF TRUTH…Sketch a graph of h(x), j(x) and the ratio function.
h(x)
j(x)
Let’s give it another go. Consider 𝑘(𝑥) = (𝑥 2 + 6𝑥 + 9) ∙ (𝑥 − 2)
For what value(s) of x is the function equal to zero?
For what value(s) of x is the function infinitely close to zero but positive?
For what value(s) of x is the function infinitely close to zero but negative?
For what value(s) of x is the function infinitely large but negative?
For what value(s) of x is the function infinitely large but positive?
It might be helpful to sketch k(x)…
1
What does all of this mean for the reciprocal function? Consider the behavior of 𝑘(𝑥) at each of
the x values identified above. When you are ready provide a graph of the reciprocal function.
Consider m(x) = x and the ratio function
𝑚(𝑥)
𝑘(𝑥)
MOMENT OF TRUTH…Sketch a graph of m(x), k(x) and the ratio function.
m(x)
k(x)
Compare and contrast the ratio function
𝑚(𝑥)
𝑘(𝑥)
1
to the reciprocal function 𝑘(𝑥). Offer an explanation
as to why the graphs of the functions are similar and different.