Problem 61 - 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. $$g(x)=2 x^{2}(x+3)$$

Step-by-Step Solution

Answer

The function with the same end behavior is \(y=2x^2\). The x-intercepts are \(x=-3\) and \(x=0\), and the y-intercept is \(y=0\). The function is positive when \(x > 0\) and \(-3 < x < 0\), and negative when \(x < -3\). See finished graph for a visual representation.

Step 1: Expand the given polynomial

The polynomial \(g(x) = 2x^2 (x+3)\) can be written as \(g(x) = 2x^3 + 6x^2\). Expanding the polynomial simplifies the equation and assists in understanding its behavior.

Step 2: Find a function with the same end behavior

The end behavior of a polynomial is determined by examining the highest degree term, in this case \(2x^3\). So, the function \(y = 2x^2\) has the same end behavior as the given function.

Step 3: Determine the x and y intercepts

For the x-intercept(s), the equation is set to equal zero and solved for x. Therefore, \(g(x) = 2x^3 + 6x^2 = 0\) implies \(x = 0\) or \(x = -3\). The y-intercept is found by evaluating the function at \(x = 0\). Plugging in \(x = 0\) into \(g(x)\), we get \(y = 0\).

Step 4: Determine intervals where value of function is positive or negative

The function is positive where the output \(g(x)\) is greater than zero. Solve the inequality \(2x^3 + 6x^2 > 0\) to find \(x > 0\) and \(-3 < x < 0\). The function is negative where the output \(g(x)\) is less than zero, which gives the interval \(x < -3\).

Step 5: Sketch a graph of function

Plot the x and y intercepts on a graph, sketch the function being positive and negative as per the intervals determined in the previous steps and keeping in mind the end behavior. The function must closely resemble a \(x^3\) curve, due to the leading coefficient.

Key Concepts

End Behavior of Polynomials

Understanding the end behavior of a polynomial is crucial for sketching its graph. Essentially, end behavior describes how the polynomial behaves as the input values become either very large or very small. Let's focus on the highest degree term, because it has the greatest impact on the graph as the value of x increases or decreases without bound.

For the given function, the highest degree term is the cubic term, which means our polynomial is of the form \(y = ax^3\). Since the leading coefficient (the number before \(x^3\)), which is 2, is positive, the end behavior of our graph will mimic that of \(y = x^3\), with tails heading off to infinity in the positive direction of the y-axis as x becomes positive infinity, and in the negative direction as x becomes negative infinity. This is because any term with an odd exponent and a positive leading coefficient will show this type of end behavior.

X-intercepts of Polynomial

An x-intercept of a graph is a point where the graph crosses the x-axis, which corresponds to a root of the polynomial equation. To find these, we look for values of x that make the polynomial equal to zero. In simpler terms, we're finding the 'zeros' of our function.

With our function \(g(x) = 2x^3 + 6x^2\), setting \(g(x)=0\) gives us \(x = -3\) and \(x = 0\) after factoring out \(2x^2\). These are the x-intercepts of our polynomial - where the graph will cross or touch the x-axis.

Y-intercepts of Polynomial

The y-intercept is the point where the graph crosses the y-axis, and it's found by evaluating the function at \(x = 0\). Since polynomials are functions, they can only have one y-intercept. This is due to the fact that functions pass the vertical line test which checks that a function is indeed a function and not a relation.

For our polynomial \(g(x)\), plugging in \(x = 0\) yields \(g(0) = 0\). Therefore, the y-intercept for our graph is at the origin, \(0,0\). This also happens to coincide with one of our x-intercepts in this case.

Intervals of Positivity in Polynomials

Intervals of positivity are where the graph is above the x-axis, indicating that the function values are positive. This is crucial for graphing because it tells us about the behavior of the function between its intercepts and informs us where the function will be above the x-axis.

For our polynomial \(g(x)\), solving \(2x^3 + 6x^2 > 0\) determines that the function is positive for \(x > 0\) and between \(x = -3\) and \(x = 0\), excluding \(x = -3\) itself if it's not a closed interval. The function will lie above the x-axis in these intervals.

Intervals of Negativity in Polynomials

Conversely, intervals of negativity indicate where the graph of the polynomial function lies below the x-axis, where the function values are negative. By finding these intervals, we can sketch the portions of the graph that dip below the x-axis.

From the inequality \(2x^3 + 6x^2 < 0\), we discern that \(g(x)\) is negative for \(x < -3\). This tells us the part of the graph that will be plotted below the x-axis.

Sketching Polynomial Graphs

To sketch a graph of a polynomial, we combine all the previously mentioned characteristics. Start by plotting the x and y intercepts, then use the knowledge of end behavior to determine how the graph approaches infinity. Next, plot the intervals of positivity and negativity, ensuring the graph crosses the x-axis at the intercepts and stays above/below the axis in the correct intervals.

Using the information we've gathered, we know the graph of \(g(x) = 2x^3 + 6x^2\) touches or crosses the x-axis at \(x = -3\) and \(x = 0\), starts below the x-axis for very negative x values, crosses upwards through \(x = -3\), remains above the axis until it once more touches at \(x = 0\), and finally continues upwards as x-value increases.