Verified Answer :
- The polynomial \(g(x) = x^3 + 5x^2 - 12x - 36\) is a cubic function with end behavior as \(x \to \pm\infty\), \(g(x) \to \pm\infty\).
- The y-intercept is \((0, -36)\) and the x-intercepts are at \(x = -6, -2, 3\).
- Local maximum and minimum points occur at approximately \((-4.27, 28.55)\) and \((0.94, -42.03)\).
- These characteristics define the shape of the graph, allowing for its identification.
Explanation
- Analyzing Polynomial Features
To analyze the polynomial function \(g(x) = x^3 + 5x^2 - 12x - 36\), we first identify its key features. The function is a cubic polynomial with a positive leading coefficient (1), meaning as \(x \to \infty, g(x) \to \infty\) and as \(x \to -\infty, g(x) \to -\infty\). The y-intercept is found by evaluating \(g(0)\):
\[g(0) = 0^3 + 5(0)^2 - 12(0) - 36 = -36.\]
This indicates the graph crosses the y-axis at the point \((0, -36)\).
- Finding x-intercepts
To find the x-intercepts, we solve \(x^3 + 5x^2 - 12x - 36 = 0\). Testing potential rational roots, we find that \(x = 3\) is a solution:
\[g(3) = 3^3 + 5(3^2) - 12(3) - 36 = 27 + 45 - 36 - 36 = 0.\]
Dividing the polynomial by \((x-3)\), we get \(x^2 + 8x + 12 = 0\), which factors as \((x+6)(x+2) = 0\). Thus, the roots are \(x = -6, x = -2,\) and \(x = 3\). The graph intersects the x-axis at \((-6, 0), (-2, 0),\) and \((3, 0)\).
- Determining Local Extrema
Next, we determine the local extrema by finding the roots of the derivative \(g'(x) = 3x^2 + 10x - 12\). Setting \(g'(x) = 0\), we use the quadratic formula:
\[x = \frac{-10 \pm \sqrt{100 - 4(3)(-12)}}{2(3)} = \frac{-10 \pm \sqrt{244}}{6} = \frac{-5 \pm \sqrt{61}}{3}.\]
Evaluating these values gives \(x \approx -4.27\) and \(x \approx 0.94\). The local maximum is at approximately \((-4.27, 28.55)\) and the local minimum is at approximately \((0.94, -42.03)\). Combining all these characteristics, we can identify the graph that exhibits these intercepts and turning points.
Examples
Understanding how to graph polynomials like \(g(x) = x^3 + 5x^2 - 12x - 36\) is essential for analyzing trends in data that aren't strictly linear. For instance, in economics, a cubic model might represent how profit changes with production volume: positive regions show profitability, intercepts show the break-even points, and extrema highlight production levels that maximize profit or minimize losses. Mastering these skills allows for precise prediction and decision-making in various analytical fields.