Asked by walter white
Describe how the graph of y= 2 times 3^x-1 compares to its parent function
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GPT-5 mini
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Do you mean y = 2·3^x − 1 or y = 2·3^{x−1}? The notation is ambiguous. Briefly, both possibilities compared to the parent y = 3^x:
1) If y = 2·3^x − 1
- Vertical stretch by factor 2 (points are twice as far from the x-axis).
- Vertical shift down 1 unit.
- Horizontal asymptote moves from y = 0 to y = −1.
- y-intercept: (0, 1) because 2·3^0 − 1 = 1.
- x-intercept: x = log_3(1/2) ≈ −0.6309.
2) If y = 2·3^{x−1}
- Vertical stretch by factor 2.
- Horizontal shift right 1 unit (because of x − 1 inside the exponent).
- Horizontal asymptote remains y = 0.
- y-intercept: (0, 2/3) because 2·3^{−1} = 2/3.
- No finite vertical shift.
(If you meant y = 2·3^{x−1} − 1, both the right shift and the down shift apply: stretch by 2, right 1, down 1, asymptote y = −1.) Which form did you intend?
1) If y = 2·3^x − 1
- Vertical stretch by factor 2 (points are twice as far from the x-axis).
- Vertical shift down 1 unit.
- Horizontal asymptote moves from y = 0 to y = −1.
- y-intercept: (0, 1) because 2·3^0 − 1 = 1.
- x-intercept: x = log_3(1/2) ≈ −0.6309.
2) If y = 2·3^{x−1}
- Vertical stretch by factor 2.
- Horizontal shift right 1 unit (because of x − 1 inside the exponent).
- Horizontal asymptote remains y = 0.
- y-intercept: (0, 2/3) because 2·3^{−1} = 2/3.
- No finite vertical shift.
(If you meant y = 2·3^{x−1} − 1, both the right shift and the down shift apply: stretch by 2, right 1, down 1, asymptote y = −1.) Which form did you intend?
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