on ramps pre cal

Question

if a function f(x) has values f(4)=6 and f(8)=18, use what you have learned about function patterns to find f(16)=if f(x) is:

a) linear function: f(16)=

b) power function: f(16)=

c) exponential function: f(16)=

d) logarithmic function: f(16)=

Answer 1

1) Linear Function:

m = (18 - 6)/(8 - 4) = 12/4 = 3

y - 6 = 3(x - 4)

y = 3x - 6

f(16) = 3(16) - 6 = 48 - 6 = 42

2) Power Function:

y = axⁿ

f(4) = 6 → 6 = a(4)ⁿ...............(1)

f(8) = 18 → 18 = a(8)ⁿ...........(2)

Taking (2) and dividing it by (1), we get

3 = 2ⁿ

ln3 = ln2ⁿ

ln3 = n(ln2)

n = ln3/ln2 = log₂3

Solving for a with equation (1)

6 = a(4)^log₂³

6 = a(2)^2log₂³ = a(2)^log₂⁹ = a(9)

a = 6/9 = 2/3

y = 2/3 x^log₂³

f(16) = 2/3 (16)^log₂³ = 2/3 (2)^4log₂³ = 2/3 (2)^log₂⁸¹ = 2/3 (81) = 54

3) Exponential function:

y = ab^x

f(4) = 6 → 6 = ab⁴............(1)

f(8) = 18 → 18 = ab⁸........(2)

Same technique as above divide eqn (2) by eqn (1)

3 = b⁴

b = (3)^1/4

6 = a((3)^1/4)⁴ = a(3)

a = 2

y = 2(3)^x/4

f(16) = 2(3)^16/4 = 2(3)⁴ = 162

4) Logarithmic

y = a + b(lnx)

f(4) = 6 → 6 = a + b(ln4).........(1)

f(8) = 16 → 16 = a + b(ln8).....(2)

(2) - (1)

10 = b(ln8 - ln4) = b(ln(8/4)) = b(ln2)

b = 10/ln2

6 = a + 10/(ln2)[ln4] = a + 10(2ln2)/(ln2) = a + 20

a = -14

y = -14 + (10/ln2)lnx

f(16) = -14 + (10/ln2)(ln16) = -14 + (10/ln2)(4ln2) = -14 + 40 = 26

Hope this helps

Mr. K