Connect a factorised higher-degree rule and its graph practice questions | Mathematical Methods (Year 12, VCE Units 3 & 4, Applications of differentiation, Tangents, normals and curve sketching)

Question 1 of 8

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Consider the polynomial rule below. Let \(P(x)\) be given by:

P(x)=-4(x-\frac{1}{3})^{2}(x+\frac{2}{3})^{2}

An exact calibration point is \(P(-\frac{1}{3})=-\frac{16}{81}\). Use the graph window \(-3\le x\le 3\) and \(-385\le y\le 385\) for the graph selection

(a)5 marks

Use the stated factorised rule to complete the table of ordered x-intercepts, their multiplicities and the vertical intercept

\(x\)-intercept 1 in increasing order multiplicity at \(x\)-intercept 1 \(x\)-intercept 2 in increasing order multiplicity at \(x\)-intercept 2 vertical intercept \(P(0)\)

(b)1 mark

Which statement describes the intercept behaviour?

An odd repeated factor is treated as an ordinary crossing. At x=-2/3, the graph touches the horizontal axis and turns. At x=1/3, the graph touches the horizontal axis and turns

At x=-2/3, the graph touches the horizontal axis and turns. At x=1/3, the graph touches the horizontal axis and turns

The parity rule is reversed. At x=-2/3, the graph crosses the horizontal axis. At x=1/3, the graph crosses the horizontal axis

Repeated factors are treated as touch points. At x=-2/3, the graph touches the horizontal axis and turns. At x=1/3, the graph touches the horizontal axis and turns

Which end behaviour is consistent with P?

The graph rises on both the left and the right

The graph falls on the left and rises on the right

The graph rises on the left and falls on the right

The graph falls on both the left and the right

(d)1 mark

Which graph is consistent with the rule and its features?