Question

which graph represents a function with a growth factor of 5?

Explanation:

Step1: Recall exponential function form

An exponential growth function has the form \( y = a \cdot b^x \), where \( b \) is the growth factor (\( b > 1 \)), and \( a \) is the initial value (\( x = 0 \) gives \( y = a \)).

Step2: Analyze each graph's points

• For a growth factor of 5, when \( x = 1 \), \( y = a \cdot 5^1 = 5a \); when \( x = 0 \), \( y = a \); when \( x = -1 \), \( y = a \cdot 5^{-1}=\frac{a}{5} \).

Check the first graph (top - left):

• Points: \( (-1, 0.125) \), \( (0, 0.625) \), \( (1, 3.125) \)? Wait, no, the given points: \( (-1, 0.125) \), \( (0, 0.5) \), \( (1, 2) \)? Wait, recalculate. Wait, if \( x = 0 \), \( y = a \), so \( a = 0.5 \) (from \( (0, 0.5) \)). Then at \( x = 1 \), \( y = 0.5 \cdot 5 = 2.5 \)? But the graph has \( (1, 2) \)? Wait, no, maybe I misread. Wait, let's check the third graph (top - right, third one? Wait, the first top - left: points \( (-1, 0.125) \), \( (0, 0.5) \), \( (1, 2.5) \)? Wait, no, the user's graph: first top - left has \( (0, 0.5) \), \( (1, 2) \)? Wait, no, let's do the ratio. From \( x = 0 \) to \( x = 1 \), \( y \) should multiply by 5. So if \( x = 0 \), \( y = a \), \( x = 1 \), \( y = 5a \), \( x = -1 \), \( y = a/5 \).

Check the third graph (top - right, the one with \( (0, 0.6) \)? No, wait the third graph (top - right) has \( (-1, 0.06) \), \( (0, 0.3) \)? Wait, no, the third graph (top - right) has \( (-1, 0.06) \), \( (0, 0.3) \)? Wait, no, let's check the first top - left: \( (-1, 0.125) \), \( (0, 0.5) \), \( (1, 2.5) \)? Wait, the first graph's \( (0, 0.5) \), \( (1, 2) \)? No, maybe the first graph (top - left) has \( (0, 0.5) \), \( (1, 2.5) \)? Wait, no, the user's first graph (top - left) has \( (0, 0.5) \), \( (1, 2) \)? Wait, no, let's calculate the ratio between \( x = 0 \) and \( x = 1 \). For the first graph: \( y(1)/y(0)=2/0.5 = 4 \)? No. Wait the second graph (top - middle): \( (0, 0.2) \), \( (1, 1) \), ratio \( 1/0.2 = 5 \)! Wait, \( x = 0 \), \( y = 0.2 \); \( x = 1 \), \( y = 1 \). \( 1/0.2 = 5 \). Then \( x = -1 \), \( y = 0.2/5 = 0.04 \), which matches the point \( (-1, 0.04) \) in the second graph (top - middle). Yes! So the second graph (top - middle) has \( x = -1 \): \( 0.04 \), \( x = 0 \): \( 0.2 \), \( x = 1 \): \( 1 \). Let's check: \( 0.2 \cdot 5^1 = 1 \) (correct for \( x = 1 \)), \( 0.2 \cdot 5^{-1}=0.04 \) (correct for \( x = -1 \)). So this graph has a growth factor of 5.

Wait, no, wait the second graph (top - middle) has \( (0, 0.2) \), \( (1, 1) \), \( (-1, 0.04) \). So \( 0.2 \times 5 = 1 \) (x=1), \( 0.2 / 5 = 0.04 \) (x=-1). So the growth factor is 5.

Step3: Confirm the growth factor

The growth factor \( b \) is the ratio of \( y(x + 1)/y(x) \). For the second graph (top - middle), \( y(1)/y(0)=1/0.2 = 5 \), \( y(0)/y(-1)=0.2/0.04 = 5 \). So the growth factor is 5.

Answer:

The Middle Graph (top - middle graph)