Worked Example: Comprehensive Problem Solutions

Calculus Readiness · Axiom Academy

Example 1 Algebra Meets Functions Let . Describe the graph: find the domain, any holes, intercepts, and end behavior. Domain: The denominator cannot be zero, so . The domain is all reals except x = 2 . Factor the numerator: x^2 - 4 = (x - 2)(x + 2) , so . Cancel: for , f(x) = x + 2 . There is a hole at x = 2 , not a vertical asymptote. Hole location: evaluate the simplified form at x = 2 : 2 + 2 = 4 . The hole is at (2, 4) . Intercepts: the y -intercept is f(0) = 0 + 2 = 2 . The x -intercept solves x + 2 = 0 , so x = -2 . End behavior: as , — it behaves like a line. Example 2 Trig + Transformations Graph . Identify the amplitude, period, phase shift, and vertical shift. Rewrite in standard form: factor out of the argument to get . Identify parameters: A = 2 , , , D = 1 . Amplitude: |A| = 2 . The graph oscillates 2 above and below the midline. Period: . One full cycle takes units. Phase shift: to the right (this is C/B , not C ). Vertical shift: the midline is y = 1 , so the graph oscillates between y = -1 and y = 3 . Example 3 Logs and Exponentials A population grows according to , where t is in years. (a) Find the population after 9 years. (b) Find the time for the population to reach 4000 . Divide both sides by 500 : 2^ t/3 = 8 . Recognize 8 = 2^3 : 2^ t/3 = 2^3 , so , giving t = 9 . Alternative using logs: , so t = 9 — the same answer. Example 4 A Limit Using Every Tool

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