Problem 64 - Precalculus : Building Concepts and Connections

Question

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$g(x)=-(x-2)(3 x-1)\left(x^{2}+1\right)$$

Step-by-Step Solution

Answer

The polynomial function is \(g(x) = -3x^4 +6x^3 -x^2 +2x\). This is equivalent to \(y = -3x^2\) and hence has similar end behavior. The \(x\)-intercepts are \(x = 2\) and \(x = 1/3\), and the \(y\)-intercept is \(y = 2\). The function is positive between \(x = 1/3\) and \(x = 2\), and negative for \(x < 1/3\) and \(x > 2\). Use this information to draw the graph of the function.

Step 1: Expansion and form

Expand \(g(x)\) to obtain \(g(x) = -3x^4 +6x^3 -x^2 +2x\). This is of the form \(y=cx^2\), where \(c\) is the coefficient of the highest degree term. So, the function with the same end behavior is \(y = -3x^2\).

Step 2: The x and y intercepts

Set \(x\) to zero to find the \(y\)-intercept, which results in \(g(0) = 2\). For the \(x\)-intercepts, set \(y\) or \(g(x)\) to zero and solve. This gives us \(x = 2\) and \(x = 1/3\). So, the \(x\)-intercepts are \(x = 2\) and \(x = 1/3\), and the \(y\)-intercept is \(y = 2\).

Step 3: Positive Intervals

The function will change signs at the \(x\)-intercepts. To find the interval where the function is positive, pick a test point from each interval divided by the \(x\)-intercepts and substitute it in the simplified polynomial to check where \(g(x)\) is positive. Using this strategy and considering the intervals \(-\infty < x < 1/3\), \(1/3 < x < 2\) and \(2 < x < \infty\), where \(x\) equals \(0\), \(1\), and \(3\), we find that the function is positive between \(x = 1/3\) and \(x = 2\).

Step 4: Negative Intervals

To find the interval where the function is negative, we also analyze the signs at the test points. The function is negative for \(x < 1/3\) and \(x > 2\).

Step 5: Graphing the Function

Use all the previously found information to sketch a graph of the function. You should note where the function passes through the \(x\) and \(y\) axis and the intervals where the function is positive and negative.

Key Concepts

End Behavior

End behavior of a polynomial function describes how the function behaves as the value of \(x\) approaches positive or negative infinity. This is largely determined by the highest degree term present in the polynomial. In our given polynomial function \(g(x) = -(x-2)(3x-1)(x^2 + 1)\), when expanded, it becomes \(-3x^4 + 6x^3 - x^2 + 2x\). The leading term, \(-3x^4\), dictates the end behavior. To summarize:

• If the leading coefficient is negative, as \(x\) goes to infinity the function will decrease, and as \(x\) goes to negative infinity, the function will increase, mirroring the downward opening of a negative quadratic function like \(y = -3x^2\).
• For positive leading coefficients, the opposite occurs; the function increases in both directions.

Therefore, \(g(x)\) has the same end behavior as \(y = -3x^2\): it decreases without bound as \(x\) approaches positive or negative infinity.

Intercepts

Intercepts are critical to understand where a graph crosses the axes. For the polynomial \(g(x) = -(x-2)(3x-1)(x^2+1)\), we need both the \(x\)-intercepts and the \(y\)-intercept.

• X-Intercepts: These occur where \(g(x) = 0\). Solving \(-(x-2)(3x-1)(x^2 + 1) = 0\) gives us the solutions: \(x=2\) and \(x=\frac{1}{3}\), as the quadratic factor no real roots.
• Y-Intercept: This is found by evaluating \(g(x)\) when \(x = 0\). Substituting \(0\) into the expanded form, we get \(g(0) = 2\), indicating the graph crosses the \(y\)-axis at \(y = 2\).

Understanding where the function intersects the axes helps us sketch and visualize the polynomial function accurately.

Positive and Negative Intervals

Identifying when a function is positive or negative across different intervals helps in sketching its graph more accurately. These intervals are divided by the \(x\)-intercepts.For \(g(x)\):

• Positive Intervals: To determine when \(g(x)\) is greater than \(0\), test points in the intervals between intercepts, such as checking \(x = 1\) in \((\frac{1}{3}, 2)\). Here, \(g(x) > 0\), so the function is positive.
• Negative Intervals: Similarly, for \(g(x) < 0\), test regions to the left of \(\frac{1}{3}\) and right of \(2\), finding \(g(x) < 0\) in both \((-\infty, \frac{1}{3})\) and \((2, \infty)\).

Understanding these intervals means recognizing where the curve lies above or below the \(x\)-axis, crucial for graphing.

Graphing Functions

Graphing polynomial functions involves merging all the learned elements: end behavior, intercepts, and positive and negative intervals.Steps to graph \(g(x)\):

• Start by marking intercepts: plot \(x\)-intercepts at \(x=2\) and \(x=\frac{1}{3}\), and the \(y\)-intercept at \(y=2\).
• Use the end behavior information; since \(g(x)\) behaves like \(y = -3x^2\), it decreases as \(x\) tends to both positive and negative infinity.
• Draw the curve through the intercepts, noting the intervals: the curve is above the \(x\)-axis between \((\frac{1}{3}, 2)\) and below it in other intervals.

With careful observation of all these details, you can sketch a coherent graph of the polynomial, accurately representing its characteristics.