1. **State the problem:** We are given points on an exponential function: $(-1, \frac{1}{7})$, $(0, 1)$, $(1, 7)$, and $(2, 49)$. We want to show that the function grows by equal factors over intervals of length 1. 2. **Recall the property of exponential functions:** An exponential function grows by a constant factor over equal intervals. This means the ratio of function values at $x+1$ and $x$ is constant. 3. **Calculate the ratio over the interval from $x = -1$ to $x = 0$:** $$\text{Ratio} = \frac{f(0)}{f(-1)} = \frac{1}{\frac{1}{7}} = 1 \times \frac{7}{1} = 7$$ 4. **Calculate the ratio over the interval from $x = 0$ to $x = 1$:** $$\text{Ratio} = \frac{f(1)}{f(0)} = \frac{7}{1} = 7$$ 5. **Calculate the ratio over the interval from $x = 1$ to $x = 2$:** $$\text{Ratio} = \frac{f(2)}{f(1)} = \frac{49}{7} = 7$$ 6. **Interpretation:** The simplified ratio, which represents the growth factor over each interval of length 1, is 7. 7. **Conclusion:** Since the ratio is the same (7) over each interval of length 1, the function grows by equal factors over intervals of length 1, confirming it is exponential. **Final answer:** The growth factor over each interval of length 1 is $7$.