Problem

Analyze \(f(x)~=~3~\cdot~2^x~-~5\). State \(a,~b,~h,~\text{and}~k\); classify growth or decay and any reflection; give the intercepts, asymptote, domain, range, and monotonic behavior; and compute exact anchors at \(x~=~-1,~0,~\text{and}~1\).

Prompt coordinate grid scaled for the stated exponential equation; the curve, anchors, and asymptote are withheld.

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Parameters: a = 3, b = 2, h = 0, k = −5.
Class: decay because 0 < b < 1; no x-axis reflection.
Features: y-intercept (0, −2); asymptote y = −5; domain all real numbers; range y > −5; increasing.
Anchors: (−1, −7/2), (0, −2), (1, 1).

Parameters: a = 3, b = 2, h = 0, k = −5.
Class: growth because b > 1; no x-axis reflection.
Features: y-intercept (0, −2); asymptote y = −5; domain all real numbers; range y > −5; decreasing.
Anchors: (−1, −7/2), (0, −2), (1, 1).

Parameters: a = −3, b = 2, h = 0, k = −5.
Class: growth base reflected across the x-axis.
Features: y-intercept (0, −8); asymptote y = −5; domain all real numbers; range y < −5; decreasing.
Anchors: (−1, −13/2), (0, −8), (1, −11).

Parameters: a = 3, b = 2, h = 0, k = −5.
Class: growth because b > 1; no x-axis reflection.
Features: y-intercept (0, −2); asymptote y = −5; domain all real numbers; range y > −5; increasing.
Anchors: (−1, −7/2), (0, −2), (1, 1).

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Analyze \(f(x)~=~3~\cdot~2^x~-~5\). State \(a,~b,~h,~\text{and}~k\); classify growth or decay and any reflection; give the intercepts, asymptote, domain, range, and monotonic behavior; and compute exact anchors at \(x~=~-1,~0,~\text{and}~1\).

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