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Calculus IChapter 1: ReviewSection 1-10

Common Graphs — Question 1

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Question 1

Sketch the graph of the function below. Identify the domain, range, intercepts, asymptotes, and any transformations applied to a basic parent function. f(x)=−1x−2+3f(x) = -\frac{1}{x - 2} + 3

Instructions: Analyze the function using transformations of the reciprocal function f(x)=1xf(x) = \frac{1}{x}.

Original worksheet page 1: question and worked solution for 1-10-001
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Question 1 - Solution

We are given: f(x)=−1x−2+3f(x) = -\frac{1}{x - 2} + 3

Parent Function: 1x\frac{1}{x}

Transformations:

  • Horizontal shift right by 2

  • Reflection over x-axis (due to the negative sign)

  • Vertical shift up by 3

Domain: (−∞,2)∪(2,∞)(-\infty, 2) \cup (2, \infty)

Range: (−∞,3)∪(3,∞)(-\infty, 3) \cup (3, \infty)

Vertical Asymptote: x=2x = 2

Horizontal Asymptote: y=3y = 3

Y-intercept: f(0)=72f(0) = \frac{7}{2}

X-intercept: f(x)=0⇒x=73f(x) = 0 \Rightarrow x = \frac{7}{3}

See the diagram in the original worksheet below.

Original worksheet page 2: question and worked solution for 1-10-001

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