Question

NO LINKS!! Please help me with these problems. Exponential.

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Answer 1

`\textsf{1.} \quad f(x)=10400(0.25)^x`

(-1, 41600) and (3, 162.5)

`\textsf{2.} \quad g(x)=1.8(2.4)^x`

3. Sarah is incorrect.

Explanation:

`\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$f(x)=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}`

Question 1

The y-intercept is when x = 0.

Therefore, from inspection of the given table, the y-intercept is 10400:

`\implies a=10400`

Substitute point (1, 2600) and a = 10400 into the exponential function and solve for b:

`\implies 2600=10400b^1`

`\implies 2600=10400b`

`\implies b=(2600)/(10400)`

`\implies b=0.25`

Therefore, the equation of the function is:

`\boxed{f(x)=10400(0.25)^x}`

Find the value of x when f(x) = 41600:

`\begin{aligned} f(x)&=41600\\\implies 10400(0.25)^x&=41600\\(0.25)^x &=(41600)/(10400)\\(0.25)^x &=4\\\ln (0.25)^x &=\ln 4\\x \ln (0.25)&=\ln 4\\x&=(\ln 4)/(\ln (0.25))\\x&=-1\end{aligned}`

Find the value of f(x) when x = 3:

`\begin{aligned}x=3 \implies f(3) & =10400(0.25)^3\\& =10400(0.015625)\\& = 162.5\end{aligned}`

Therefore, the completed table is:

`\begin{array}\cline{1-7} \vphantom{\frac12}x & -1 & 0 & 1 & 3 & 4 & 6\\\cline{1-7} \vphantom{\frac12}f(x) &41600 &10400 &2600 &162.5 &40.625 & 2.5390625\\\cline{1-7} \end{array}`

Question 2

Given points:

• (1, 4.32)
• (4, 59.71968)

Substitute the given points into the exponential formula g(x) = abˣ

`\implies ab=4.32`

`\implies ab^4=59.71968`

To find b, divide the equations:

`\implies (ab^4)/(ab)=(59.71968)/(4.32)`

`\implies b^3=13.824`

`\implies b=\sqrt[3]{13.824}`

`\implies b=2.4`

Substitute one of the points and the found value of b into the equation and solve for a:

`\implies 2.4a=4.32`

`\implies a=(4.32)/(2.4)`

`\implies a=1.8`

Therefore, the equation of the function is:

`\boxed{g(x)=1.8(2.4)^x}`

Question 3

Sarah is incorrect. For an exponential function in the form a · bˣ:

• If a > 0 and b > 1 then it is an increasing function.
• If a > 0 and 0 < b < 1 then it is a decreasing function.

Answer 2

1. f(x) = 10400·(1/4)^x; table values: x = -1, f(x) = 162.5
2. g(x) = 1.8·2.4^x
3. Incorrect for b < 1.

Explanation:

You want to write exponential functions through the given points.

1. Table

The exponential function ...

f(x) = a·b^x

will have the values ...

• f(0) = a
• f(1) = ab ⇒ b = f(1)/f(0)

Using these relations we can write the function from the table values:

f(x) = 10400·(2600/10400)^x

f(x) = 10400·(1/4)^x

Then your table is ...

`\begin{array}c\cline{1-7}\vphantom{(b)/(g)}x&-1&0&1&3&4&6\\\cline{1-7}\vphantom{(b)/(g)}f(x)&41600&10400&2600&162.5&40.625&2.5390625\\\cline{1-7}\end{array}`

2. Points

Using the given points in the exponential function form, we have ...

4.32 = a·b^1

59.71968 = a·b^4

Dividing the second equation by the first, we find ...

59.71968/4.32 = b^3 ⇒ b = 2.4

Using this in the first equation, we get ...

4.32 = a·2.4 ⇒ a = 1.8

The equation of the function is ...

g(x) = 1.8·2.4^x

3. Increasing

The exponential function ...

y = a·b^x . . . . . . a > 0, b > 0

will be increasing when the growth factor (b) is greater than 1. When 'b' is less than 1, the function will be decreasing.

Sarah's belief is incorrect.