<h2>Functions of the Form \(y = ax^{3} + bx^{2} + cx + d\)</h2>
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<h2>Optional Investigation: The effects of \(a\) on a cubic function</h2>
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<p>Complete the table below and plot the graphs of \(f(x)\) and \(g(x)\) on the same system of axes.</p>
<p>Be careful to choose a suitable scale for the \(y\)-axis.</p>

\[f(x) = 2x^{3} - 5x^{2} - 14x + 8 \qquad \qquad g(x) = -2x^{3} + 5x^{2} + 14x - 8\]

<table>
<tbody>
<tr>
<td>\(x\)</td>
<td>\(-\text{3}\)</td>
<td>\(-\text{2}\)</td>
<td>\(-\text{1}\)</td>
<td>\(\text{0}\)</td>
<td>\(\text{1}\)</td>
<td>\(\text{2}\)</td>
<td>\(\text{3}\)</td>
<td>\(\text{4}\)</td>
<td>\(\text{5}\)</td>
</tr>
<tr>
<td>\(f(x)\)</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td>\(-\text{25}\)</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>\(g(x)\)</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td>\(\text{25}\)</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<p><strong>The effects of \(a\) on a cubic graph</strong></p>
<img alt="d24d40e4e1002a1f21b8bb7ad891e0e8.png" src="https://data.ulearngo.com/assets/76a6fef0-5c84-4f3a-bdaa-431ec241f761"/></div>
</div>

77e14597-4d58-4399-be3f-b1bb5c28bc54

Functions of the Form \(y = ax^{3} + bx^{2} + cx + d\)

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

Mathematics

Explore the principles of calculus including limits, notation, differentiation, second derivatives, optimization problems, and more, in order to gain a solid understanding of mathematical functions and graphing.


\(x\)	\(-\text{3}\)	\(-\text{2}\)	\(-\text{1}\)	\(\text{0}\)	\(\text{1}\)	\(\text{2}\)	\(\text{3}\)	\(\text{4}\)	\(\text{5}\)
\(f(x)\)							\(-\text{25}\)
\(g(x)\)							\(\text{25}\)

Functions of the Form \(y = ax^{3} + bx^{2} + cx + d\)

Functions of the Form \(y = ax^{3} + bx^{2} + cx + d\)

Optional Investigation: The effects of \(a\) on a cubic function

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