1. **State the problem:** We need to match each exponential function with its corresponding graph based on the function's behavior. 2. **Recall the general form of an exponential function:** $$y = a \cdot b^t$$ where $a$ is the initial value (y-intercept) and $b$ is the base that determines growth or decay. - If $b > 1$, the function is increasing (exponential growth). - If $0 < b < 1$, the function is decreasing (exponential decay). 3. **Analyze each function:** - $y = 60(1.4)^t$: $a=60$, $b=1.4 > 1$ (increasing, moderately steep) - $y = 90(0.93)^t$: $a=90$, $b=0.93 < 1$ (decreasing, slow decay) - $y = 90(1.03)^t$: $a=90$, $b=1.03 > 1$ (increasing, very slow growth) - $y = 30(1.6)^t$: $a=30$, $b=1.6 > 1$ (increasing, steep) - $y = 30(1.4)^t$: $a=30$, $b=1.4 > 1$ (increasing, moderately steep) - $y = 90(0.83)^t$: $a=90$, $b=0.83 < 1$ (decreasing, faster decay) 4. **Match with graphs:** - Curve I (red): increasing sharply from low — likely $y=30(1.6)^t$ (steepest growth) - Curve II (green): increasing less steep than I — likely $y=60(1.4)^t$ - Curve III (blue): increasing less steep than II — likely $y=30(1.4)^t$ - Curve IV (gray): straight line with positive slope — not exponential, so no match - Curve V (black): horizontal or nearly horizontal — no exponential growth or decay, no match - Curve VI (orange): decreasing exponential starting high — likely $y=90(0.83)^t$ (faster decay) 5. **Remaining functions:** - $y=90(1.03)^t$ is increasing very slowly, so it fits best with Curve III or II if we consider subtle differences. - $y=90(0.93)^t$ is decreasing slowly, so it fits with a less steep decay, possibly Curve VI if it is less steep than $0.83^t$. 6. **Final matching:** - I (red): $y=30(1.6)^t$ - II (green): $y=60(1.4)^t$ - III (blue): $y=90(1.03)^t$ - VI (orange): $y=90(0.83)^t$ Note: Curves IV and V are not exponential functions.