Question

algebra 2 with statistics
u6 l1 exploring and graphing rational functions
day 1- exploring rational graphs
i. reciprocal graphs:

1. using a graphing calculator, graph ( f(x)=\frac{1}{x} ) using a standard window.

a. as ( x ) is approaching positive infinity, what does it look like ( y ) is approaching?
b. as ( x ) is approaching negative infinity, what does it look like ( y ) is approaching?
c. as ( x ) is approaching zero from the right, what does it look like ( y ) is approaching?
d. as ( x ) is approaching zero from the left, what does it look like ( y ) is approaching?

1. keeping the same graph, go to table set and change it to:

table setup
tblstart = 0

Explanation:

Step1: Analyze \(x\to+\infty\)

As \(x\) gets larger and larger (\(x\to+\infty\)), for \(y = \frac{1}{x}\), when \(x = 1000\), \(y=\frac{1}{1000}=0.001\); when \(x = 10000\), \(y=\frac{1}{10000}=0.0001\). As \(x\) approaches positive infinity (\(x\to+\infty\)), \(y=\frac{1}{x}\) approaches \(0\). Mathematically, \(\lim_{x\to+\infty}\frac{1}{x}=0\)

Step2: Analyze \(x\to-\infty\)

As \(x\) gets more and more negative (\(x\to-\infty\)), for \(y=\frac{1}{x}\), when \(x=- 1000\), \(y =-\frac{1}{1000}=-0.001\); when \(x=-10000\), \(y=-\frac{1}{10000}=-0.0001\). As \(x\) approaches negative infinity (\(x\to-\infty\)), \(y = \frac{1}{x}\) approaches \(0\). Mathematically, \(\lim_{x\to-\infty}\frac{1}{x}=0\)

Step3: Analyze \(x\to0^{+}\)

As \(x\) approaches \(0\) from the right (\(x\to0^{+}\), \(x>0\) and \(x\) is very small), for example, when \(x = 0.1\), \(y=\frac{1}{0.1}=10\); when \(x=0.01\), \(y=\frac{1}{0.01} = 100\). As \(x\to0^{+}\), \(y=\frac{1}{x}\) approaches \(+\infty\). Mathematically, \(\lim_{x\to0^{+}}\frac{1}{x}=+\infty\)

Step4: Analyze \(x\to0^{-}\)

As \(x\) approaches \(0\) from the left (\(x\to0^{-}\), \(x < 0\) and \(|x|\) is very small), for example, when \(x=-0.1\), \(y=\frac{1}{-0.1}=-10\); when \(x=-0.01\), \(y=\frac{1}{-0.01}=-100\). As \(x\to0^{-}\), \(y=\frac{1}{x}\) approaches \(-\infty\). Mathematically, \(\lim_{x\to0^{-}}\frac{1}{x}=-\infty\)

Answer:

a. \(y\) is approaching \(0\)
b. \(y\) is approaching \(0\)
c. \(y\) is approaching \(+\infty\)
d. \(y\) is approaching \(-\infty\)