1. The problem asks us to graph the equation \( y = 2^{x-1} - 3 \). 2. This is an exponential function with base 2, shifted right by 1 unit and down by 3 units. 3. To graph it, calculate some values: - For \( x=0 \), \( y=2^{0-1} - 3 = 2^{-1} - 3 = \frac{1}{2} - 3 = -2.5 \). - For \( x=1 \), \( y=2^{1-1} - 3 = 2^0 - 3 = 1 - 3 = -2 \). - For \( x=2 \), \( y=2^{2-1} - 3 = 2^1 - 3 = 2 - 3 = -1 \). - For \( x=3 \), \( y=2^{3-1} - 3 = 2^2 - 3 = 4 - 3 = 1 \). 4. Plot these points and draw a smooth curve through them. The curve will rise exponentially, starting near \( y=-3 \) and increasing as \( x \) increases. 5. Next, solve for \( y \) in the equation \( (\frac{1}{27})^{y-4} = 9^{3y+1} \). 6. Rewrite bases as powers of 3 since \( 27 = 3^3 \) and \( 9 = 3^2 \): $$ \left(3^{-3}\right)^{y-4} = \left(3^2\right)^{3y+1} $$ 7. Simplify exponents: $$ 3^{-3(y-4)} = 3^{2(3y+1)} $$ 8. Since the bases are equal, set exponents equal: $$ -3(y-4) = 2(3y+1) $$ 9. Distribute: $$ -3y + 12 = 6y + 2 $$ 10. Collect like terms: $$ 12 - 2 = 6y + 3y $$ $$ 10 = 9y $$ 11. Solve for \( y \): $$ y = \frac{10}{9} $$ Final answers: - Graph the function \( y=2^{x-1} -3 \) as shown. - The solution to the equation is \( y = \frac{10}{9} \).