Final Answer:
A → red (steepest increasing)
B → green (decreasing, gentle slope)
C → orange (decreasing, steeper than B)
D → blue (increasing, less steep than A)
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Explanation:
We need to match each equation to its graph by looking at two things:
1. Does the graph go
up (increasing) or
down (decreasing) as x gets bigger?
2. How
steep is it?
Recall:
- If the base $ a > 1 $, then $ y = a^x $ increases (goes up).
- If $ 0 < a < 1 $, then $ y = a^x $ decreases (goes down).
- Also, $ y = a^{-x} = \left(\frac{1}{a}\right)^x $, so if $ a > 1 $, then $ a^{-x} $ is decreasing.
Now check each option:
A: $ y = 6^x $
Base = 6 (>1) → increasing.
Big base → grows fast → steepest upward curve → matches the
red curve.
B: $ y = 1.5^{-x} $
Rewrite: $ y = \left(\frac{1}{1.5}\right)^x = \left(\frac{2}{3}\right)^x $.
Base = $ \frac{2}{3} \approx 0.67 $ (<1) → decreasing.
But not too small → gentle decrease → matches the
green curve.
C: $ y = 0.2^x $
Base = 0.2 (<1) → decreasing.
Very small base → drops quickly → steeper downward curve → matches the
orange curve.
D: $ y = 2.5^x $
Base = 2.5 (>1) → increasing.
Less than 6, so slower growth than A → less steep upward curve → matches the
blue curve.
So:
- Red = A
- Green = B
- Orange = C
- Blue = D
Parent Tip: Review the logic above to help your child master the concept of graphing exponential equations worksheet.
