Question

the table of ordered pairs (x, y) gives an exponential function. write an equation for the function.\
\

x	y	\
---	---	\
-1	\\(\frac{1}{20}\\)	\
0	\\(\frac{1}{2}\\)	\
1	5	\
2	50

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( y = ab^x \), where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base.

Step2: Find the value of \( a \)

When \( x = 0 \), \( y=\frac{1}{2}\). Substitute \( x = 0 \) and \( y=\frac{1}{2}\) into \( y = ab^x \):
\( \frac{1}{2}=ab^0 \)
Since \( b^0 = 1 \) for any \( b
eq0 \), we get \( a=\frac{1}{2} \).

Step3: Find the value of \( b \)

Now we know \( a=\frac{1}{2} \), use another point to find \( b \). Let's use \( x = 1 \), \( y = 5 \). Substitute into \( y=\frac{1}{2}b^x \):
\( 5=\frac{1}{2}b^1 \)
Multiply both sides by 2: \( 10 = b \). We can check with other points. For \( x = 2 \), \( y=\frac{1}{2}(10)^2=\frac{1}{2}\times100 = 50 \), which matches. For \( x=- 1 \), \( y=\frac{1}{2}(10)^{-1}=\frac{1}{2}\times\frac{1}{10}=\frac{1}{20} \), which also matches.

Step4: Write the equation

Substitute \( a=\frac{1}{2} \) and \( b = 10 \) into \( y = ab^x \), we get \( y=\frac{1}{2}(10)^x \).

Answer:

\( y=\frac{1}{2}\times10^{x} \)