Graphing y = 2e⁻⁰·⁵ˣ + 3

Algebra 2 · Axiom Academy

EXAMPLE Graphing y=2e^ -0.5x +3 Identify every transformation of a decaying exponential, then find its asymptote, key points, and end behavior to graph it. Analyze and graph y = 2e^ -0.5x + 3 : identify the parent function and every transformation, find the horizontal asymptote and key points, and determine the domain, range, and end behavior. Reflect across the y -axis — negative sign in the exponent Horizontal stretch by factor 2 — the exponent's coefficient has magnitude The curve passes through (-2, 8.44) , (0, 5) , and (2, 3.74) , decreasing everywhere and flattening toward the dashed asymptote y=3 as x increases, while rising without bound to the left. Excellent work! You've completed a full transformation analysis of an exponential decay function. Here's the process that always works: Start with the parent function: every exponential builds on y=e^x . Read the exponent first: the sign and size of k control reflection and horizontal stretch or compression. Track the outer multiplier and shift: a stretches vertically, and c moves the asymptote to y=c . Find the y -intercept by substituting x=0 . Determine behavior: the sign of k tells you increasing vs. decreasing; the domain is always all real numbers, and the range is bounded by the asymptote. Complete answer: y=2e^ -0.5x +3 has asymptote y=3 , y -intercept (0,5) , is decreasing everywhere, with domain and range .

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