Question

(1 point)
\\( y = 10 \cdot (1.05)^x \\)
\\( y = 1.05 \cdot (10)^x \\)
\\( y = -10 \cdot (1.05)^x \\)
\\( y = 10 \cdot -(1.05)^x \\)

Explanation:

Step1: Analyze the general form of exponential function

The general form of an exponential function is \( y = a \cdot b^x \), where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base.

Step2: Find the initial value (\(a\))

From the graph, when \( x = 0 \), \( y = 10 \). Substituting \( x = 0 \) into the general form \( y = a \cdot b^x \), we get \( y = a \cdot b^0 = a \cdot 1 = a \). So \( a = 10 \).

Step3: Find the base (\(b\))

We also know the point \( (1, 10.5) \). Substitute \( x = 1 \), \( y = 10.5 \) and \( a = 10 \) into the equation \( y = a \cdot b^x \):
\( 10.5 = 10 \cdot b^1 \)
Divide both sides by 10: \( b = \frac{10.5}{10} = 1.05 \)

Step4: Determine the function

Substituting \( a = 10 \) and \( b = 1.05 \) into the general form, we get \( y = 10 \cdot (1.05)^x \).

Answer:

\( y = 10 \cdot (1.05)^x \) (the first option)