Question

which graph represents a function with a growth factor of 5? (graphs shown with points (-1, 0.125), (0, 0.25), (1, 0.5) on one and another partially visible)

Explanation:

To determine which graph represents a function with a growth factor of 5, we analyze the general form of an exponential growth function, \( y = a \cdot b^x \), where \( b \) is the growth factor.

Step 1: Recall the exponential growth model

The standard form of an exponential growth function is \( y = a \cdot b^x \), where:

• \( a \) = initial value (when \( x = 0 \)),
• \( b \) = growth factor (\( b > 1 \) for growth).

Step 2: Analyze the first graph (provided points)

For the first graph, we have points:

• \( (0, 0.25) \): When \( x = 0 \), \( y = 0.25 \), so \( a = 0.25 \).
• \( (1, 0.5) \): Substitute \( x = 1 \), \( y = 0.5 \), \( a = 0.25 \) into \( y = a \cdot b^x \):

\( 0.5 = 0.25 \cdot b^1 \)
Solve for \( b \): \( b = \frac{0.5}{0.25} = 2 \).
The growth factor here is \( 2 \), not \( 5 \).

Step 3: Analyze the second graph (implied, likely with growth factor 5)

For a growth factor of \( 5 \), the function would be \( y = a \cdot 5^x \). Let’s verify with a point (e.g., \( x = 1 \)):

• If \( a = 1 \) (initial value at \( x = 0 \)), then \( y = 1 \cdot 5^1 = 5 \).
• For \( x = 2 \), \( y = 1 \cdot 5^2 = 25 \), which would show rapid growth (steeper curve than the first graph).

Step 4: Compare growth factors

The first graph has a growth factor of \( 2 \). The second graph (with steeper growth) likely represents a function with \( b = 5 \), as exponential functions with larger \( b \) grow faster.

Answer:

The graph with the steeper curve (second graph, implied) represents a function with a growth factor of 5. (Note: Since the second graph’s details are partially visible, but the first graph has a growth factor of 2, the second graph is the correct one.)