Verified Answer :
- Identify the key features of the polynomial \(f(x) = x^3 + 3x^2 - 10x - 24\): y-intercept at \((0, -24)\) and x-intercepts at \(-4, -2,\) and \(3\).
- Locate the local extrema by finding critical points of the derivative \(f'(x) = 3x^2 + 6x - 10\), giving a local maximum at \(\approx (-3.08, 6.04)\) and a local minimum at \(\approx (1.08, -30.04)\).
- Recognize that the end behavior (positive leading coefficient) and these calculated points define the curve's shape.
- The correct graph reflects these intercepts and turning points.
Explanation
- Analysis of the Polynomial Function
To understand the shape of the graph of the polynomial function \(f(x) = x^3 + 3x^2 - 10x - 24\), we first analyze its key features, including its y-intercept, x-intercepts, and critical points. This function is a cubic polynomial with a positive leading coefficient, implying that as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to -\infty\).
- Finding Intercepts
The y-intercept is the point where the graph crosses the y-axis, occurring at \(x=0\). Evaluating the function at \(x=0\):
\[f(0) = (0)^3 + 3(0)^2 - 10(0) - 24 = -24.\]
Thus, the y-intercept is \((0, -24)\).
Next, we find the x-intercepts by solving \(f(x) = 0\). Testing integer factors of the constant term \(-24\), we find that \(f(-2) = (-2)^3 + 3(-2)^2 - 10(-2) - 24 = -8 + 12 + 20 - 24 = 0\). Dividing the polynomial by \((x+2)\) gives the quadratic factor \(x^2 + x - 12\), which factors as \((x+4)(x-3)\). Thus, the roots are \(x = -4, x = -2,\) and \(x = 3\). The graph intersects the x-axis at \((-4, 0), (-2, 0),\) and \((3, 0)\).
- Identifying Extrema
The local extrema occur where the derivative \(f'(x) = 0\). The derivative of the function is:
\[f'(x) = 3x^2 + 6x - 10.\]
Setting \(f'(x) = 0\) and solving using the quadratic formula:
\[x = \frac{-6 \pm \sqrt{36 - 4(3)(-10)}}{6} = \frac{-6 \pm \sqrt{156}}{6} = -1 \pm \frac{\sqrt{39}}{3} \approx -1 \pm 2.08.\]
The critical points are approximately \(x_1 \approx -3.08\) and \(x_2 \approx 1.08\).
Evaluating the function at these points gives the local maximum at approximately \((-3.08, 6.04)\) and the local minimum at approximately \((1.08, -30.04)\).
- Conclusion
Combining these findings: the graph passes through \((-4, 0), (-2, 0),\) and \((3, 0)\) on the x-axis, has a y-intercept at \((0, -24)\), rises to a local maximum at \((-3.08, 6.04)\), descends to a local minimum at \((1.08, -30.04)\), and continues upwards for \(x > 1.08\). This set of characteristics uniquely defines the cubic curve. The correct graph is the one that passes through these specific points and exhibits this specific turning behavior.
Examples
Polynomial graphing is essential in fields like engineering and economics. For instance, if a company models its profit over time using a cubic function, the x-intercepts identify the "break-even" points where profit is zero, while the local extrema identify the peak profit and the point of maximum loss. Understanding how to graph these functions allows for informed business decisions and strategic planning.