1. The problem is to understand and analyze the exponential function $y=4^x$ and compare it with other exponential functions like $y=2^x$ and $y=2 \cdot 2^x$. 2. The general form of an exponential function is $y=a^x$ where $a>0$ and $a \neq 1$. 3. Important rules: - When $a>1$, the function shows exponential growth. - The function passes through the point $(0,1)$ because $a^0=1$. 4. For $y=4^x$, the base $a=4$ which is greater than 1, so the function grows exponentially. 5. At $x=0$, $y=4^0=1$. 6. At $x=7$, $y=4^7=16384$, which is very large, explaining the steep rise. 7. Comparing $y=4^x$ with $y=2^x$ and $y=2 \cdot 2^x$: - $y=2^x$ grows slower than $y=4^x$ because 2 is smaller than 4. - $y=2 \cdot 2^x = 2^{x+1}$ grows similarly to $y=2^x$ but shifted upwards by a factor of 2. 8. The graph shows exponential growth curves with $y=4^x$ growing fastest among the three. Final answer: The function $y=4^x$ is an exponential growth function with base 4, passing through $(0,1)$ and increasing steeply as $x$ increases.