To graph the exponential function \( y = 0.5(6)^x \) and identify the y-intercept, we follow these steps:
Step 1: Identify the y-intercept
The y-intercept of a function occurs when \( x = 0 \). Plugging \( x = 0 \) into the function:
\[ y = 0.5(6)^0 \] \[ y = 0.5 \times 1 = 0.5 \]
Y-Intercept:
The y-intercept is \( (0, 0.5) \).
Step 2: Graph the function
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Calculate Values for different \( x \):
- For \( x = -2 \): \[ y = 0.5(6)^{-2} = 0.5 \times \frac{1}{36} \approx 0.0139 \]
- For \( x = -1 \): \[ y = 0.5(6)^{-1} = 0.5 \times \frac{1}{6} \approx 0.0833 \]
- For \( x = 0 \) (already calculated): \[ y = 0.5 \]
- For \( x = 1 \): \[ y = 0.5(6)^{1} = 0.5 \times 6 = 3 \]
- For \( x = 2 \): \[ y = 0.5(6)^{2} = 0.5 \times 36 = 18 \]
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Plot the Points:
- \( (-2, 0.0139) \)
- \( (-1, 0.0833) \)
- \( (0, 0.5) \)
- \( (1, 3) \)
- \( (2, 18) \)
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Draw the Curve:
- Start from roughly \( (-2, 0.0139) \) and move upwards through the points to the right, forming a steep curve as \( x \) increases.
Summary of the Information:
- Y-Intercept: \( (0, 0.5) \)
- Graph: Will have a steep increase as \( x \) becomes positive and very small values as \( x \) becomes negative.
When graphing, the function will show an exponential growth curve starting from near zero for negative \( x \) values and increasing rapidly for positive \( x \) values.