Question

the graph represents the function $f(x) = 10(2)^x$
how would the graph change if the $b$-value in the equation is decreased but remains greater than 1?
choose two correct answers.
the graph will increase at a faster rate
the $y$-values will each be less than their corresponding $x$-values
the graph will begin at a lower point on the $y$-axis
the graph will increase at a slower rate
the $y$-values will continue to increase as $x$ increases

Explanation:

1. For an exponential growth function \(f(x)=a(b)^x\) where \(b>1\), decreasing \(b\) (while keeping it >1) reduces the growth rate, so the graph increases slower.
2. A smaller \(b\) means for each \(x\), \(b^x\) is smaller, so \(y=a(b)^x\) will be less than the original \(y\)-value for the same \(x\).
3. All exponential growth functions with \(b>1\) still have \(y\)-values that increase as \(x\) increases, regardless of the size of \(b\) (as long as it stays above 1).

Answer:

• The y-values will each be less than their corresponding x values
• The graph will increase at a slower rate
• The y-values will continue to increase as x increases