Question

Kaylee needed to write the equation of an exponential function from points on the graph of the function. To determine the value of b, Kaylee chose the ordered pairs (1, 6) and (3, 48) and divided 48 by 6. She determined that the value of b was 8. What error did Kaylee make? Complete the explanation.

Answer 1

One

Explanation:

General form of an exponential function:
`y=ab^x`

where a is the initial value, b is the growth/decay factor, and x is the independent variable

If b > 1 the function will grow (increase). If 0 < b < 1 then the function will decay (decrease)

The value Kaylee found is not the constant ratio of successive values. She should use ordered pairs with x-values that differ by 1

This is how Kaylee computed the value of b, which was incorrect as the solution of 8 is for
`b^2`:

at (1, 6):
`ab^1=6`

at (3, 48)
`ab^3=48`

`\implies (ab^3)/(ab^1)=(48)/(6)\\\\\implies b^2=8\\\\\implies b=\pm√(8) =2√(2)`

(b is positive since the function is increasing)

However, if Kaylee used the ordered pairs with a difference of one, e.g. x = 1 and x = 2, then she would have computed the correct value of b:

at (1, 6):
`ab^1=6`

at x = 2:
`ab^2=y`

`\implies (ab^2)/(ab^1)=(y)/(6)\\\\\implies b=(y)/(6)`

Answer 2

• 1

Explanation:

The exponent function is:

• f(x) = a*bˣ

You can find the value of b if you divide the consecutive values of f(x).

• f(x + 1) = a*bˣ⁺¹

• f(x + 1)/f(x) = a*bˣ⁺¹/a*bˣ = bˣ⁺¹⁻ˣ = b

The missing number in the explanation is 1.

Kaylee gets:

• f(3) / f(1) = 48/6 = 8 but it is the value of b² not b.