To match each equation with its corresponding graph, we need to analyze the characteristics of exponential functions:

Key Observations:

1. Exponential Growth (Base > 1): Functions like $y = 7^x$ and $y = 11^x$ will show rapid growth, rising steeply from left to right. The larger the base, the steeper the curve.

2. Exponential Decay (0 < Base < 1): Functions like $y = 0.7^x$ and $y = \left(\frac{1}{8}\right)^x$ will show decay, falling from left to right. The smaller the base, the steeper the decay.

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Matching the Graphs:

• (A) $y = 7^x$: This shows strong exponential growth. Since f is below k on the left and above k on the right, it grows slower than k. So, f = $y = 7^x$.

• (B) $y = 0.7^x$: This decays slowly because the base is close to 1. Since h is above g on the left and below on the right, and decays slower, h = $y = 0.7^x$.

• (C) $y = 11^x$: This grows faster than $7^x$, so it must be steeper. Since k rises faster than f, k = $y = 11^x$.

• (D) $y = \left(\frac{1}{8}\right)^x$: This decays faster because $\frac{1}{8}$ is much smaller than 1. Since g decays faster than h, g = $y = \left(\frac{1}{8}\right)^x$.

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Final Matching:

• (A) $y = 7^x$ → f
• (B) $y = 0.7^x$ → h
• (C) $y = 11^x$ → k
• (D) $y = \left(\frac{1}{8}\right)^x$ → g

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Would you like me to explain any part of this in more detail?

Related Questions:

1. How does the base of an exponential function affect its growth or decay rate?
2. What is the difference between exponential growth and exponential decay graphs?
3. How do exponential functions behave as $x$ approaches negative infinity?
4. Can you determine the y-intercept for these exponential functions?
5. How would the graphs change if we added constants to the equations?

Tip:

When comparing exponential functions, focus on the base—larger bases grow faster, and bases between 0 and 1 decay.