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Not your average video:
Interactive Problems: Solve problems alongside the video with step-by-step guidance and detailed solutions.
Exam Preparation: Complete unit reviews designed for final exam preparation with all key concepts covered systematically.
Expert Teaching: High-quality instruction from Perplex co-founder James Mullen with clear explanations, worked examples, and exam tips.
Transformations & asymptotes
Watch comprehensive video reviews for Transformations & asymptotes, designed for final exam preparation. Each video includes integrated problems you can solve alongside detailed solutions.
Not your average video:
Interactive Problems: Solve problems alongside the video with step-by-step guidance and detailed solutions.
Exam Preparation: Complete unit reviews designed for final exam preparation with all key concepts covered systematically.
Expert Teaching: High-quality instruction from Perplex co-founder James Mullen with clear explanations, worked examples, and exam tips.
Not your average video:
Interactive Problems: Solve problems alongside the video with step-by-step guidance and detailed solutions.
Exam Preparation: Complete unit reviews designed for final exam preparation with all key concepts covered systematically.
Expert Teaching: High-quality instruction from Perplex co-founder James Mullen with clear explanations, worked examples, and exam tips.
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AHL 2.13
When a rational function is of the form
There will be vertical asymptotes when the quadratic cx2+dx+e=0.
The horizontal asymptote will simply by y=0 since cx2 dominates ax when x is very large.
Additionally, there will be an x-intercept at x=−ab, when the numerator changes sign.
Note that if the numerator and denominator share a root (ie x=−ab is a root of the denominator), then there will only be one asymptote and a "hole" on the x-axis. This has never showed up on exams.
Example
Sketch the graph of y=x2−x−6x−1.
The denominator is (x−3)(x+2), which is zero when x=3 and x=−2.
The numerator is 0 when x=1 (negative to the left and positive to the right). When x<−2, the function is negative (plug in −3)
So we plot the asymptotes x=3 and x=−2, and start tracing from the negative side of the x-axis. At each vertical asymptote the sign will change, and we have an x-intercept at 1.
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AHL 2.13
When a rational function is of the form
There will be vertical asymptotes when the quadratic cx2+dx+e=0.
The horizontal asymptote will simply by y=0 since cx2 dominates ax when x is very large.
Additionally, there will be an x-intercept at x=−ab, when the numerator changes sign.
Note that if the numerator and denominator share a root (ie x=−ab is a root of the denominator), then there will only be one asymptote and a "hole" on the x-axis. This has never showed up on exams.
Example
Sketch the graph of y=x2−x−6x−1.
The denominator is (x−3)(x+2), which is zero when x=3 and x=−2.
The numerator is 0 when x=1 (negative to the left and positive to the right). When x<−2, the function is negative (plug in −3)
So we plot the asymptotes x=3 and x=−2, and start tracing from the negative side of the x-axis. At each vertical asymptote the sign will change, and we have an x-intercept at 1.
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