Question

complete the table for the function $y = 2(5)^x$.

x	y
-1	\square
0	\square
1	\square
2	\square

now, graph the function.
plot two points to graph the function.
(graph with x-axis from -7 to 7 and y-axis from -14 to 14, grid lines.)

Explanation:

Step1: Substitute \( x=-1 \) into \( y = 2(5)^{x} \)

We know that \( a^{-n}=\frac{1}{a^{n}} \), so \( 5^{-1}=\frac{1}{5} \). Then \( y = 2\times5^{-1}=2\times\frac{1}{5}=\frac{2}{5}=0.4 \).

Step2: Substitute \( x = 0 \) into \( y = 2(5)^{x} \)

Any non - zero number to the power of 0 is 1, so \( 5^{0}=1 \). Then \( y = 2\times5^{0}=2\times1 = 2 \).

Step3: Substitute \( x = 1 \) into \( y = 2(5)^{x} \)

\( 5^{1}=5 \), so \( y=2\times5^{1}=2\times5 = 10 \).

Step4: Substitute \( x = 2 \) into \( y = 2(5)^{x} \)

\( 5^{2}=25 \), so \( y=2\times5^{2}=2\times25 = 50 \).

For graphing, we can choose two points, for example, when \( x = 0,y = 2 \) (the point is \( (0,2) \)) and when \( x = 1,y = 10 \) (the point is \( (1,10) \)). We can plot these two points on the coordinate plane and then draw the curve of the exponential function \( y = 2(5)^{x} \) through them.

Answer:

For \( x = -1 \): \( y=\frac{2}{5} \) (or \( 0.4 \)); For \( x = 0 \): \( y = 2 \); For \( x = 1 \): \( y=10 \); For \( x = 2 \): \( y = 50 \)