<p>The variables \(x\) and \(y\) are connected by the equation \(y = x^3 - 3x^2 - 1\).</p>
<p>The table below shows some corresponding values of \(x\) and \(y\).</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">\(x\)</th><th style="text-align: center; --cell-justify: center">\(-2\)</th><th style="text-align: center; --cell-justify: center">\(-1\)</th><th style="text-align: center; --cell-justify: center">\(0\)</th><th style="text-align: center; --cell-justify: center">\(1\)</th><th style="text-align: center; --cell-justify: center">\(2\)</th><th style="text-align: center; --cell-justify: center">\(3\)</th><th style="text-align: center; --cell-justify: center">\(4\)</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(y\)</td><td style="text-align: center; --cell-justify: center">\(-21\)</td><td style="text-align: center; --cell-justify: center">\(-5\)</td><td style="text-align: center; --cell-justify: center">\(-1\)</td><td style="text-align: center; --cell-justify: center">\(-3\)</td><td style="text-align: center; --cell-justify: center">\(a\)</td><td style="text-align: center; --cell-justify: center">\(-1\)</td><td style="text-align: center; --cell-justify: center">\(15\)</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Calculate the value of \(a\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Taking \(2\) cm to represent \(1\) unit on the \(x\)-axis and \(2\) cm to represent \(5\) units on the \(y\)-axis, draw the graph of \(y = x^3 - 3x^2 - 1\) for the range of \(-2 \leq x \leq 4\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Using your graph, find the value of \(x\) when&nbsp;\(y = 10\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Using your graph, solve the equation</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">\(x^3 - 3x^2 - 2 = 0\),</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(x^3 - 3x^2 - 2x + 2 = 0\).</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">By drawing a tangent, find the gradient of the curve where&nbsp;\(x = 2.5\).</span></div></li></ol>

The variables \(x\) and \(y\) are connected by the equation \(y = x^3 - 3x^2 - 1\).

The table below shows some corresponding values of \(x\) and \(y\).

Taking \(2\) cm to represent \(1\) unit on the \(x\)-axis and \(2\) cm to represent \(5\) units on the \(y\)-axis, draw the graph of \(y = x^3 - 3x^2 - 1\) for the range of \(-2 \leq x \leq 4\).

Functions and Graphs - Graphical Solution and Gradient of the Curve at a Point

Question 2d453e84