4. write an exponential function $y = ab^{x}$ whose graph passes through the given points.

Explanation:

Step1: Substitute (1, 0.5) into formula

Substitute $x=1$, $y=0.5$ into $y=ab^x$: $0.5 = ab^1 \implies a = \frac{0.5}{b}$

Step2: Substitute (4, 4) and $a$

Substitute $x=4$, $y=4$, $a=\frac{0.5}{b}$ into $y=ab^x$: $4 = \frac{0.5}{b} \cdot b^4$ Simplify: $4 = 0.5b^3$

Step3: Solve for $b$

Isolate $b^3$: $b^3 = \frac{4}{0.5} = 8$ Take cube root: $b = \sqrt[3]{8} = 2$

Step4: Solve for $a$

Substitute $b=2$ into $a=\frac{0.5}{b}$: $a = \frac{0.5}{2} = 0.25 = \frac{1}{4}$

Step5: Write final function

Substitute $a=\frac{1}{4}$ and $b=2$ into $y=ab^x$: $y = \frac{1}{4} \cdot 2^x$ Simplify (optional): $y = 2^{x-2}$

Answer:

$y = \frac{1}{4} \cdot 2^x$ (or equivalent $y=2^{x-2}$)