Question
5. The exponential function \( f(x) \) passes through the points \( (0,150) \) and \( (1,90) \). (3) Find a formula. Show work. Your formula should begin with " \( y= \) ".
Ask by Cole Herrera. in the United States
Feb 19,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The formula for the exponential function is \( y = 150(0.6)^x \).
Solution
An exponential function has the form
y = ab^x.
Step 1: Use the point (0, 150):
When x = 0, y = ab^0 = a · 1 = a. So, a = 150.
Step 2: Use the point (1, 90):
When x = 1, y = 150 · b = 90.
Solve for b:
150 · b = 90 ⟹ b = 90/150 = 0.6.
Thus, the formula for the exponential function is
y = 150(0.6)^x.
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Simplify this solution Mind Expander
To find the formula for the exponential function \( f(x) = ab^x \) that passes through the points \( (0, 150) \) and \( (1, 90) \), we can use these points to set up equations. 1. From the point \( (0, 150) \), we substitute into the equation: \[ f(0) = ab^0 = a = 150. \] 2. Substituting the point \( (1, 90) \): \[ f(1) = ab^1 = 150b = 90. \] To find \( b \), we solve for \( b \): \[ b = \frac{90}{150} = \frac{3}{5}. \] So, we have \( a = 150 \) and \( b = \frac{3}{5} \). Finally, the formula is: \[ y = 150 \left( \frac{3}{5} \right)^x. \]
