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VICSpecialist MathematicsQuick questions

Unit 3: Functions, relations and graphs

Quick questions on Rational functions and graphing: VCE Specialist Mathematics Unit 3

8short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What are domain and vertical asymptotes?
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The function is undefined where Q(x)=0Q(x) = 0. If PP and QQ have no common factor there, the line x=ax = a is a vertical asymptote and the graph shoots to ±∞\pm\infty on each side. If PP and QQ share the factor (xβˆ’a)(x - a), there is instead a hole (point discontinuity) at x=ax = a.
What are intercepts?
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The yy-intercept is f(0)f(0) (when 00 is in the domain). The xx-intercepts are the zeros of the numerator that are not also zeros of the denominator, that is, the solutions of P(x)=0P(x) = 0 with Q(x)β‰ 0Q(x) \neq 0.
What is vertical asymptote?
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The denominator is zero at x=1x = 1, and the numerator there is 12+1=2β‰ 01^2 + 1 = 2 \neq 0, so x=1x = 1 is a vertical asymptote (no hole).
What is oblique asymptote?
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Since deg⁑P=2\deg P = 2 is one more than deg⁑Q=1\deg Q = 1, divide. Polynomial division gives
What is behaviour near x=1x = 1?
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Just to the right (x=1+x = 1^+), 2xβˆ’1β†’+∞\frac{2}{x-1} \to +\infty; just to the left (x=1βˆ’x = 1^-), 2xβˆ’1β†’βˆ’βˆž\frac{2}{x-1} \to -\infty. So the curve rises to +∞+\infty on the right branch and falls to βˆ’βˆž-\infty on the left branch, hugging y=x+1y = x + 1 far from x=1x = 1.
What is q1?
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State all asymptotes of f(x)=3xβˆ’6x+1f(x) = \dfrac{3x - 6}{x + 1}. [2 marks]
What is q2?
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Find the oblique asymptote of f(x)=2x2+xxβˆ’1f(x) = \dfrac{2x^2 + x}{x - 1}. [2 marks]
What is q3?
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Describe how y=∣x2βˆ’1∣y = |x^2 - 1| differs from y=x2βˆ’1y = x^2 - 1. [2 marks]

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