- Set \(g(x) = 0\) to find the \(x\)-intercepts by solving \((x + 1)(x - 4)(x - 6) = 0\), which gives \(x = -1, 4, 6\).
- Evaluate \(g(0)\) to find the \(y\)-intercept, which yields \(g(0) = 24\).
- Match these results to the first option provided.
- Final Answer: \(\boxed{\text{x-intercepts: -1, 4, 6; y-intercept: 24}}\)
Explanation
- Analysis of Intercepts
To find the \(x\)-intercepts and the \(y\)-intercept of the function \(g(x) = (x + 1)(x^2 - 10x + 24)\), we begin by examining the definition of intercepts for any function \(g(x)\). The \(x\)-intercepts occur where the function value is zero, i.e., \(g(x) = 0\). The \(y\)-intercept occurs where the input variable is zero, i.e., at \(x = 0\).
- Finding x-intercepts
To find the \(x\)-intercepts, we solve the equation \(g(x) = (x + 1)(x^2 - 10x + 24) = 0\). This product equals zero if at least one of its factors is zero. Thus, we have either \(x + 1 = 0\) or \(x^2 - 10x + 24 = 0\). From the first equation, we immediately find \(x = -1\). For the second part, we look to factor the quadratic expression \(x^2 - 10x + 24\). We need two numbers that multiply to \(24\) and sum to \(-10\). These numbers are \(-4\) and \(-6\), so the expression factors as \((x - 4)(x - 6)\). Setting this to zero, we get \(x = 4\) and \(x = 6\). Therefore, the \(x\)-intercepts are \(-1, 4,\) and \(6\).
- Finding y-intercept and Conclusion
To find the \(y\)-intercept, we evaluate the function at \(x = 0\). Substituting \(x = 0\) into the expression for \(g(x)\), we obtain:
\[g(0) = (0 + 1)(0^2 - 10(0) + 24) = 1 \times 24 = 24.\]
Thus, the \(y\)-intercept is \(24\). Comparing our findings with the given options, we see that the correct choice is the first one.
Examples
In physics, when analyzing the motion of an object described by a polynomial path, finding the intercepts is crucial. The \(x\)-intercepts represent the moments in time (or positions) when the object passes through a specific reference point (like height zero), while the \(y\)-intercept represents its starting position. For a function like \(g(x) = (x + 1)(x^2 - 10x + 24)\), understanding where it crosses the axes allows an engineer to determine key transition points in a trajectory.