Verified Answer :
- Analyze the end-behavior: the graph goes to \(-\infty\) as \(x \to -\infty\) and to \(\infty\) as \(x \to \infty\).
- Determine the zeroes: factor the function to find \(x\)-intercepts at \(x = -4, 0, 2\).
- Assess sign intervals: identify that the function is negative on \((-\infty, -4)\) and \((0, 2)\), and positive on \((-4, 0)\) and \((2, \infty)\).
- Sketch the curve: plot the intercepts and use the determined behavior and sign intervals to draw the smooth, continuous cubic curve.
Explanation
- Analysis of the Function
The function given is \(f(x) = x^3 + 2x^2 - 8x\). This is a third-degree polynomial (a cubic function). The leading term is \(x^3\), which dictates the overall behavior of the function at very large positive and negative values of \(x\). To sketch this graph, we will identify key features such as the intercepts, intervals of positivity and negativity, and critical points.
- End-Behavior Analysis
Since the degree of the function is odd (3) and the leading coefficient is positive (1), the end-behavior is as follows:
- As \(x \to \infty\), \(f(x) \to \infty\).
- As \(x \to -\infty\), \(f(x) \to -\infty\).
This means the graph starts from the bottom left and goes up towards the top right.
- Finding Intercepts
To find the \(x\)-intercepts, we solve \(f(x) = 0\):
\[x^3 + 2x^2 - 8x = 0\]
Factoring out the common factor \(x\):
\[x(x^2 + 2x - 8) = 0\]
Now, factor the quadratic trinomial \(x^2 + 2x - 8\):
\[x(x+4)(x-2) = 0\]
The zeroes are \(x = -4\), \(x = 0\), and \(x = 2\). The \(y\)-intercept is found by evaluating \(f(0)\), which is \(f(0) = 0^3 + 2(0)^2 - 8(0) = 0\). Thus, the graph crosses the axes at \((-4, 0)\), \((0, 0)\), and \((2, 0)\).
- Intervals of Positivity and Negativity
Using the zeroes \(-4, 0,\) and \(2\), we divide the number line into four intervals and test a value in each to determine if the function is positive or negative:
- Interval \((-\infty, -4)\): Test \(x = -5 \implies f(-5) = -5(-5+4)(-5-2) = -5(-1)(-7) = -35 < 0\).
- Interval \((-4, 0)\): Test \(x = -1 \implies f(-1) = -1(-1+4)(-1-2) = -1(3)(-3) = 9 > 0\).
- Interval \((0, 2)\): Test \(x = 1 \implies f(1) = 1(1+4)(1-2) = 1(5)(-1) = -5 < 0\).
- Interval \((2, \infty)\): Test \(x = 3 \implies f(3) = 3(3+4)(3-2) = 3(7)(1) = 21 > 0\).
These sign changes confirm the graph passes through the x-axis at the roots.
- Sketching the Graph
To complete the sketch:
- Plot the x-intercepts at \((-4, 0), (0, 0),\) and \((2, 0)\).
- Draw the end-behavior: start from the bottom left and end at the top right.
- Draw smooth curves connecting the points, following the sign patterns: the graph is below the x-axis for \(x < -4\), rises above the x-axis between \(-4\) and \(0\), dips below for \(x\) between \(0\) and \(2\), and rises above for \(x > 2\).
Examples
In economics, cubic functions are often used to model production costs over different scales. A function like \(f(x) = x^3 + 2x^2 - 8x\) might represent the marginal cost or profit trend as you increase the production quantity \(x\). Identifying where the function is zero (breakeven points) and where it is positive or negative allows business owners to predict whether they will face a loss or generate profit at specific production levels, helping them make more informed decisions about resource allocation.