Interpreting Transformed Function Range: Profit Scenario Analysis

Understanding Function Range

• 🎯 The range of a function represents the set of all possible output values, corresponding to the y-values on a graph.
• 📈 In this scenario, the original function P(X) models Bernard's business profit in dollars over X months.

Analyzing P(X) Range

• 📉 The minimum profit for P(X) appears to be a loss of $3,500.
• 💰 The maximum profit for P(X) over the five months is estimated to be $11,500.
• 📊 Therefore, the range of P(X) is approximately from -$3,500 to $11,500.

Analyzing D(X) Range

• ⚙️ The transformed function D(X) is defined as P(X) - 4, indicating a shift downwards.
• 💸 Since the y-axis represents thousands of dollars, the transformation means each profit value is reduced by $4,000.
• 📉 The new minimum profit for D(X) is estimated at $7,500 ($3,500 - $4,000 is incorrect, it should be -$3,500 - $4,000 = -$7,500, but the transcript states $7,500 as the low point. Assuming the transcript meant $11,500 - $4,000 = $7,500 for the high point and the low point is also shifted down by $4,000 from -$3,500 to -$7,500. However, the transcript explicitly states the low point for D(X) is $7,500. This implies the original low point was -$3,500 and the new low point is -$7,500. The transcript states the low point is $7,500, which is $4,000 less than $11,500. This implies the original high point was $11,500 and the new high point is $7,500. The transcript also states the low point is $7,500. This is contradictory. Re-reading, the transcript says "So this is not just down by four, it's down by 4,000. So this is7500. So the low point is 7500 if I'm estimating correctly. And the high point it looks like it is it looks like it is $7500, which is $4,000 less than 11,500. So 7,500." This indicates the transcript is confused about the low point. Assuming the intent is that both the low and high points are reduced by $4,000.
• 💰 The new maximum profit for D(X) is estimated at $7,500 ($11,500 - $4,000).

Interpretation of Transformation

• ↔️ The primary interpretation is that the entire range of profits has shifted down by $4,000.
• 📉 Both the lowest and highest profit points are $4,000 lower in the transformed function D(X) compared to P(X).