Question

using your graphing calculator, investigate the graphs of the following family (group) of functions. describe any relationships within the family, including domain and range, growth or decay factors, vertical and horizontal intercepts, and asymptotes. identify the functions as increasing or decreasing.

$f(x)=10^{x},g(x)=-10^{x}$

graph $f(x)$ and $g(x)$ in the same viewing window.

all graphs are displayed $-12,12,1$ viewing window.

the domain of $f(x)$ is $(-$ decay $)$ main of $g(x)$ is $(-\infty,\infty)$. the range of $f(x)$ is $(0,\infty)$ and the range of $g(x)$ is $(-\infty,0)$.

(type your answer in int growth)

the graph of $f(x)$ has a factor of $\square$. the graph of $g(x)$ has a factor of $\square$.

(type integers or fractions.)

Explanation:

Step1: Analyze the function \(f(x) = 10^x\)

The general form of an exponential function is \(y = a\cdot b^x\). For \(f(x)=10^x\) (where \(a = 1\) and \(b=10>1\)), it is an exponential growth function. There is no vertical stretch or compression (\(a = 1\)).

Step2: Analyze the function \(g(x)=- 10^x\)

For \(g(x)=-10^x\), we can rewrite it as \(y=-1\times10^x\). Compared to the parent function \(y = 10^x\), there is a reflection about the \(x\) - axis (\(a=- 1\)).

Answer:

The graph of \(f(x)\) has a growth factor of \(1\). The graph of \(g(x)\) has a reflection factor of \(-1\).