Question

relating components of functions to their graphs

the given graph represents the function \\(f(x) = 2(5)^x\\). how will the appearance of the graph change if the \\(a\\) value in the function is decreased, but remains greater than 0?

• the graph will increase at a slower rate.
• the graph will show a decreasing, rather than increasing, function.
• the graph will show an initial value that is lower on the \\(y\\)-axis.
• the graph will increase at a constant additive rate, rather than a multiplicative rate.

Explanation:

Identify the parameters of the exponential function

$$ f(x) = a \cdot b^x = 2(5)^x $$

$$ a = 2, \quad b = 5 $$

Analyze the effect of decreasing the parameter \(a\)

$$ f(0) = a \cdot b^0 = a $$

$$ 0 < a_{\text{new}} < 2 $$

$$ f_{\text{new}}(0) < f(0) $$

Evaluate the given multiple-choice options

$$ \text{Initial value on } y\text{-axis decreases from } 2 \text{ to } a_{\text{new}} $$

Answer:

• The graph will increase at a slower rate.
• The graph will show a decreasing, rather than increasing, function.
• The graph will show an initial value that is lower on the y-axis. (Correct answer)
• The graph will increase at a constant additive rate, rather than a multiplicative rate.