Question

(see image for problem)a. the a -term:b. the y -intercept:c. the common ratio:d. the x -intercept:e. the end behavior:

Answer 1

The general form of an exponential function is expressed as

`\begin{gathered} y=ab^x\text{ ----- equation 1} \\ \text{where} \\ a\text{ is the y-intercept of the function} \\ b\text{ is the common ration of the function} \end{gathered}`

In the function

`h(x)\text{ = }-2((1)/(3))^x\text{ ------ equation 2}`

A) a-term:

In, the h(x) function, the a-term is -2.

B) the y-intercept:

The y-intercept of the function is obtained as the value of h(x), when x equals zero.

thus,

`\begin{gathered} \text{when x = 0,} \\ h(x)\text{ = -2(}(1)/(3))^0\text{ = -2}*1 \\ \Rightarrow h(x)\text{ = }-2 \end{gathered}`

thus, the y-intercept is -2.

C) the common ratio

In equation 1, b is the common ratio of the exponential function. In comparison with equation 2, we have

`b\text{ = }(1)/(3)`

Thus, the common ratio of the function is

`(1)/(3)`

D) the x-intercept:

The x-intercept of the function is obtained as the value of x when h(x) equals zero.

thus,

`\begin{gathered} \text{when h(x) = 0} \\ 0\text{ = -2(}(1)/(3))^x \\ \Rightarrow0=-2(3^(-1))^x \\ 0=3^(-x) \\ take\text{ the log of both sides,} \\ \log \text{ 0 = log}3^(-x) \\ \infty\text{ = -xlog3} \\ \Rightarrow x=\text{ }(\infty)/(-\log 3) \\ x\text{ = }\infty \\ \end{gathered}`

thus, the x-intercept is at ∞ (infinity).

E) the end behaviour:

The end behavoiur of the function is the behaviour of the h(x) function as x approaches plus infinity or negative infinity.

Thus,

`\begin{gathered} \lim _(x\to-\infty)y\text{ = }\lim _(x\to-\infty)(-2((1)/(3))^x)\text{ } \\ =-2\cdot\lim _(x\to\: -\infty\: )\mleft(\mleft((1)/(3)\mright)^x\mright) \\ =-2\cdot\infty \\ \Rightarrow-\infty \end{gathered}`
`\begin{gathered} \lim _(x\to\infty)y\text{ = }\lim _(x\to\infty)(-2((1)/(3))^x)\text{ } \\ =-2\cdot\lim _(x\to\infty\: )\mleft(\mleft((1)/(3)\mright)^x\mright) \\ =-2\cdot\: 0 \\ \Rightarrow0 \\ \end{gathered}`

thus, as x tends to negative infinity, h(x) tends to negative infinity. when x tends to positive infinity, h(x) tends to zero.

Sketch of the h(x) graph on a coordinate plane:

The sketch of the h(x) function is as shown below: