To determine which equation represents the graph that includes the points (-1, 10) and (1, 3.6), we can analyze the equations and see which one passes through those points.
- Calculate the value of y for each equation at x = -1 and x = 1.
For the first equation:
y = 6⋅(0.6)^x
- At x = -1:
\( y = 6 \cdot (0.6)^{-1} = 6 \cdot \frac{1}{0.6} = 6 \cdot \frac{10}{6} = 10 \) (correct) - At x = 1:
\( y = 6 \cdot (0.6)^1 = 6 \cdot 0.6 = 3.6 \) (correct)
For the second equation:
y = 10⋅(1.05)^x
- At x = -1:
\( y = 10 \cdot (1.05)^{-1} = 10 \cdot \frac{1}{1.05} \approx 9.52 \) (not correct) - At x = 1:
\( y = 10 \cdot (1.05)^1 = 10 \cdot 1.05 = 10.5 \) (not correct)
For the third equation:
y = 6⋅0.6^x
- At x = -1:
\( y = 6 \cdot 0.6^{-1} = 6 \cdot \frac{1}{0.6} = 10 \) (correct) - At x = 1:
\( y = 6 \cdot 0.6^1 = 6 \cdot 0.6 = 3.6 \) (correct)
For the fourth equation:
y = 6⋅(3.6)^x
- At x = -1:
\( y = 6 \cdot (3.6)^{-1} = 6 \cdot \frac{1}{3.6} \approx 1.67 \) (not correct) - At x = 1:
\( y = 6 \cdot (3.6)^1 = 6 \cdot 3.6 = 21.6 \) (not correct)
From the calculations, both the first equation \( y = 6 \cdot (0.6)^x \) and the third equation \( y = 6 \cdot 0.6^x \) give the correct values for points (-1, 10) and (1, 3.6).
However, equations (1) and (3) are essentially the same, so either would be correct. The response choices can be confusing, but the correct answer here would be:
y = 6⋅(0.6)^x or y = 6⋅0.6^x (interpreted as they are the same).