Problem 50 - 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. $$f(x)=3 x^{3}-27 x$$

Step-by-Step Solution

Answer

The function is \(y = 3x^2\). The x-intercepts are \(-3, 0, 3\) and the y-intercept is \(0\). The function is positive for \(x< -3\) and \(x> 0\), and negative for \(-3< x <0\).

Step 1: Formatting the function

To find a function of the form \(y=cx^2\) that has the same end behavior, we only need to consider the term with the highest power from the polynomial \(f(x)=3x^3-27x\). This gives us the function \(y=3x^2\).

Step 2: Finding the x-intercepts

To find the x-intercepts, we set \(f(x)\) equal to zero and solve for x: \[0=3x^3-27x\] \[0=x(3x^2-27)\] The solutions for x are \(0, \sqrt{9}\) which is \(3\), and \(-\sqrt{9}\) which is \(-3\). These are our x-intercepts.

Step 3: Finding the y-intercept

To find the y-intercept, we set \(x=0\) in the function \(f(x)\), which gives us \(0\). Therefore, the y-intercept is \(0\).

Step 4: Determining the intervals for positive function values

We need to find the intervals where \(3x^3-27x>0\). This happens when \(x< -3\) or \(x>0\). So, the function is positive in intervals \(-\infty, -3\) and \(0, \infty\).

Step 5: Determining the intervals for negative function values

We need to find the intervals where \(3x^3-27x<0\). This happens when \(-3

Step 6: Sketching the function

To sketch this function, plot the x- and y-intercepts on a graph. Because the coefficient of the highest power term \(x^3\) is positive, the graph will start down on the left and end up on the right. This sketch will cross the x-axis at \(x = -3, 0, 3\), and will be above the x-axis for \(x< -3\) and \(x> 0\), while it will be below the x-axis when \(-3

Key Concepts

End Behavior

When discussing the end behavior of a polynomial, we look at what happens to the graph as the input, or the variable \(x\), approaches infinity or negative infinity. The end behavior of polynomial functions is largely determined by the highest degree term in the polynomial.
- For the polynomial \(f(x) = 3x^3 - 27x\), the term \(3x^3\) dictates its end behavior because it is the term with the highest power.- Since the leading coefficient (3) is positive and the degree of the polynomial (3) is odd, the graph will fall to the left and rise to the right.
- This is similar to the end behavior of the quadratic function \(y = 3x^2\), which rises on both ends, but don't forget that the odd power causes the polynomial to have opposite behavior on each end.

Intercepts

Intercepts are the points where the graph of the polynomial crosses the x-axis and y-axis. These are significant as they provide key reference points for graphing.
- **X-intercepts** occur where the polynomial equals zero. For \(f(x) = 3x^3 - 27x\), setting the expression to zero results in the equation: \[ 0 = 3x(x^2 - 9) = 3x(x + 3)(x - 3) \] Therefore, the x-intercepts are \(x = -3, 0, 3\).
- The **Y-intercept** is found by setting \(x = 0\). Substituting into the function the result is \(f(0) = 0\). This means the y-intercept is also 0, at the origin (0, 0).

Intervals

Understanding the behavior of the polynomial within certain intervals can reveal where the function is positive or negative.
- To determine where the polynomial \(f(x) = 3x^3 - 27x\) is positive, solve the inequality \(3x(x + 3)(x - 3) > 0\). By considering test points in intervals defined by the x-intercepts \(-3, 0, \text{ and } 3\), you will find that the function is:

• Positive when \(x < -3\)
• Positive when \(x > 0\)

- Consequently, to determine where the function is negative, solve the inequality \(3x(x + 3)(x - 3) < 0\). The function is:

• Negative when \(-3 < x < 0\)

Graph Sketching

Graph sketching is a synthesis of all previous information, highlighting the overall shape and key points of the polynomial on a coordinate plane.
- Start by plotting the intercepts: \(-3, 0, \text{ and } 3\) on the x-axis, and the y-intercept at the origin (0, 0).
- Reflect on the end behavior derived from the leading term: the graph will fall towards negative infinity as \(x\) approaches negative infinity and rise towards positive infinity as \(x\) approaches positive infinity.
- Use the intervals to ensure the sketch shows the graph below the x-axis between \(-3\) and \(0\), and above the x-axis when \(x < -3\) and \(x > 0\).
- Finally, draw a continuous curve passing through all intercepts, respecting changes in direction as dictated by the intervals and end behavior. This builds a complete picture of the polynomial's behavior across the graph.

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Other exercises in this chapter

Problem 50

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=2 x^{5}-6 x^{3}+7 x^{2}-8$$

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Problem 50

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Problem 50

Graph the polynomial function using a graphing utility. Then (a) approximate the \(x\) -intercept(s) of the graph of the function; (b) find the intervals on whi

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Problem 51

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$h(x)=\frac{1}{x^{2}+1}$$

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Key Concepts

End Behavior

When discussing the end behavior of a polynomial, we look at what happens to the graph as the input, or the variable \(x\), approaches infinity or negative infinity. The end behavior of polynomial functions is largely determined by the highest degree term in the polynomial.
- For the polynomial \(f(x) = 3x^3 - 27x\), the term \(3x^3\) dictates its end behavior because it is the term with the highest power.- Since the leading coefficient (3) is positive and the degree of the polynomial (3) is odd, the graph will fall to the left and rise to the right.
- This is similar to the end behavior of the quadratic function \(y = 3x^2\), which rises on both ends, but don't forget that the odd power causes the polynomial to have opposite behavior on each end.

Intercepts

Intercepts are the points where the graph of the polynomial crosses the x-axis and y-axis. These are significant as they provide key reference points for graphing.
- **X-intercepts** occur where the polynomial equals zero. For \(f(x) = 3x^3 - 27x\), setting the expression to zero results in the equation: \[ 0 = 3x(x^2 - 9) = 3x(x + 3)(x - 3) \] Therefore, the x-intercepts are \(x = -3, 0, 3\).
- The **Y-intercept** is found by setting \(x = 0\). Substituting into the function the result is \(f(0) = 0\). This means the y-intercept is also 0, at the origin (0, 0).

Intervals

Understanding the behavior of the polynomial within certain intervals can reveal where the function is positive or negative.
- To determine where the polynomial \(f(x) = 3x^3 - 27x\) is positive, solve the inequality \(3x(x + 3)(x - 3) > 0\). By considering test points in intervals defined by the x-intercepts \(-3, 0, \text{ and } 3\), you will find that the function is:

• Positive when \(x < -3\)
• Positive when \(x > 0\)

- Consequently, to determine where the function is negative, solve the inequality \(3x(x + 3)(x - 3) < 0\). The function is:

• Negative when \(-3 < x < 0\)

Graph Sketching

Graph sketching is a synthesis of all previous information, highlighting the overall shape and key points of the polynomial on a coordinate plane.
- Start by plotting the intercepts: \(-3, 0, \text{ and } 3\) on the x-axis, and the y-intercept at the origin (0, 0).
- Reflect on the end behavior derived from the leading term: the graph will fall towards negative infinity as \(x\) approaches negative infinity and rise towards positive infinity as \(x\) approaches positive infinity.
- Use the intervals to ensure the sketch shows the graph below the x-axis between \(-3\) and \(0\), and above the x-axis when \(x < -3\) and \(x > 0\).
- Finally, draw a continuous curve passing through all intercepts, respecting changes in direction as dictated by the intervals and end behavior. This builds a complete picture of the polynomial's behavior across the graph.