Question

Use substitution to find an exponential function, y= a - b" that passes through
(2, 16) and (5, 1.024).

Answer 1

Answer:

$y = 4^x$

Step-by-step explanation:

Given

$(2,16)$

$(5,1,024)$

Required

Determine the exponential function $y = a.b^x$

In $(2,16)$

$x=2$ and $y = 16$

Substitute these values in $y = a.b^x$

$16 = a.b^2$ --- (1)

In $(5,1,024)$

$x = 5$ and $y = 1024$

Substitute these values in $y = a.b^x$

$1024 = a.b^5$ --- (2)

Divide (2) by (1)

$\frac{1024}{16} = \frac{a.b^5}{a.b^2}$

$\frac{1024}{16} = \frac{b^5}{b^2}$

$64 = \frac{b^5}{b^2}$

Apply second law of indices

$64 = b^{5-2$

$64 = b^3$

Express 64 as an exponent of 3

$4^3 = b^3$

$4 = b$

$b = 4$

Substitute 4 for b in (1)

$16 = a.b^2$

$16 = a * 4^2$

$16 = a * 16$

$16 = 16a$

Solve for a

$a=\frac{16}{16}$

$a = 1$

Substitute 1 for a and 4 for b in $y = a.b^x$

$y = 1 * 4^x$

$y = 4^x$

Answer 2

The exponential function is given by y = 100(0.4)ˣ

An exponential function is in the form:

y = abˣ

where y, x are variables, a is the initial value of y and b is the factor.

At point (2, 16):

16 = ab²      (1)

At point (5, 1.024):

1.024 = ab⁵      (2)

Dividing equation 2 by equation 1:

b³ = 0.064

b = 0.4

16 = a(0.4)²

a = 100

The exponential function is given by: y = 100(0.4)ˣ

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