Question

from $x = 0$ to $x = 2$, which of the following best describes the growth of the two functions? $y = 5^x$ grows over a different interval than $y = 5x$. $y = 5^x$ grows at the same rate as $y = 5x$. $y = 5x$ grows slower than $y = 5^x$. $y = 5x$ grows faster than $y = 5^x$.

Explanation:

Step1: Evaluate \( y = 5^x \) at \( x = 0 \) and \( x = 2 \)

At \( x = 0 \), \( y = 5^0 = 1 \). At \( x = 2 \), \( y = 5^2 = 25 \). The change in \( y \) is \( 25 - 1 = 24 \).

Step2: Evaluate \( y = 5x \) at \( x = 0 \) and \( x = 2 \)

At \( x = 0 \), \( y = 5(0) = 0 \). At \( x = 2 \), \( y = 5(2) = 10 \). The change in \( y \) is \( 10 - 0 = 10 \).

Step3: Compare the growth rates

Since \( 24 > 10 \) over the interval from \( x = 0 \) to \( x = 2 \), the function \( y = 5x \) (linear) grows slower than \( y = 5^x \) (exponential).

Answer:

\( y = 5x \) grows slower than \( y = 5^x \).