Plotting a Rational Function

00:00 5 marks

Consider the rational function \(f(x) = \frac{2x+3}{x-1}\).

(a) Plot the key points of the function, including intercepts and asymptotes.

(b) Discuss the domain restrictions of the function and explain how they affect the graph.

Part a

4 minutes 3 marks

Question

Plot the function \(f(x)=\frac{2x+3}{x-1}\) by determining its intercepts, vertical asymptote, and horizontal asymptote. List at least three key points on the graph.

Hint Hint

Hint

Calculate where the numerator equals zero for the x-intercept, set the denominator to zero for the vertical asymptote, and consider the leading coefficients for the horizontal asymptote.

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Complete Answer

x-intercept: \((-\frac{3}{2}, 0)\); y-intercept: \((0, -3)\); Vertical asymptote: \(x=1\); Horizontal asymptote: \(y=2\). Key points include these intercepts and points near the asymptotes, such as \((2, 7)\) and \((0.5, -8)\).

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Part b

3 minutes 2 marks

Question

Discuss the domain restrictions of \(f(x)=\frac{2x+3}{x-1}\) and explain how these restrictions are reflected in the graph of the function.

Hint Hint

Hint

Consider values that make the denominator zero and the resulting behavior of the function near these values.

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Complete Answer

The domain of \(f(x)\) is \(x \in \mathbb{R}, x
eq 1\), since \(x=1\) makes the denominator zero. This restriction is evident in the graph as a vertical asymptote at \(x=1\), indicating the function is undefined at this value and approaches infinity or negative infinity as \(x\) approaches 1.

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