Question

match each exponential function to its graph.
$s(x) = 3^x$
$t(x) = 4^x$
$k(x) = 5^x$
$v(x) = 6^x$

Explanation:

Step1: Recall exponential growth property

For exponential functions of the form $f(x)=a^x$ where $a>1$, the larger the base $a$, the faster the function increases for $x>0$, and the steeper its graph.

Step2: Evaluate at $x=1$

Calculate each function at $x=1$:

• $s(1)=3^1=3$
• $t(1)=4^1=4$
• $k(1)=5^1=5$
• $v(1)=6^1=6$

Step3: Match steepness to base

The steepest graph (highest at $x=1$) corresponds to the largest base, and the least steep to the smallest base.

1. The least steep graph (lowest at $x=1$) is $s(x)=3^x$
2. The next steepest graph is $t(x)=4^x$
3. The next steepest graph is $k(x)=5^x$
4. The steepest graph (highest at $x=1$) is $v(x)=6^x$

Answer:

1. Least steep graph: $\boldsymbol{s(x)=3^x}$
2. Next steepest graph: $\boldsymbol{t(x)=4^x}$
3. Next steepest graph: $\boldsymbol{k(x)=5^x}$
4. Steepest graph: $\boldsymbol{v(x)=6^x}$

(Note: From top-left to bottom-right, the matches are $s(x)=3^x$, $t(x)=4^x$, $k(x)=5^x$, $v(x)=6^x$ based on increasing steepness)