Question

if all the graphs above have equations with form (y = ab^x),

which graph has the largest value for (b)?
which graph has the smallest value for (b)?
which graph has the largest value for (a)?

Explanation:

⚡ Using what you learned: Graphing Exponential Functions

Step 1: Analyze the role of \( b \) (the base)

For exponential functions of the form \( y = ab^x \) with \( a > 0 \):

• If \( b > 1 \), the function represents exponential growth (the graph rises from left to right). The larger the value of \( b \), the steeper the growth curve.
• If \( 0 < b < 1 \), the function represents exponential decay (the graph falls from left to right). The smaller the value of \( b \) (closer to \( 0 \)), the steeper the decay curve.

Looking at the graph:

• Growth curves (rising from left to right): Blue (\( B \)) and Green (\( G \)). Since Blue (\( B \)) rises much faster than Green (\( G \)), it has the largest base:

$$ b_{\text{blue}} > b_{\text{green}} > 1 $$

• Decay curves (falling from left to right): Orange (\( O \)) and Purple (\( K \)). Since Orange (\( O \)) decays much faster (steeper drop) than Purple (\( K \)), it has a smaller base closer to \( 0 \):

$$ 0 < b_{\text{orange}} < b_{\text{purple}} < 1 $$

Therefore:

• The largest value for \( b \) is blue (B).
• The smallest value for \( b \) is orange (O).

Step 2: Analyze the role of \( a \) (the y-intercept)

The value of \( a \) represents the \( y \)-intercept of the function because when \( x = 0 \):

$$ y = ab^0 = a $$

Looking at the \( y \)-intercepts (where each graph crosses the vertical axis):

• The Red line (\( R \)) is a horizontal line with a constant height that is higher than the intersection point of the other curves.
• The other curves (\( B, G, O, K \)) all intersect at the same lower point on the \( y \)-axis.

Therefore, the graph with the highest \( y \)-intercept has the largest value for \( a \):

• The largest value for \( a \) is red (R).

Answer:

• Which graph has the largest value for \( b \)? blue (B)
• Which graph has the smallest value for \( b \)? orange (O)
• Which graph has the largest value for \( a \)? red (R)