Problem 57 - 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. $$f(x)=-x(2 x+1)(x-3)$$

Step-by-Step Solution

Answer

a) \(y=-2x^2\), b) x-intercepts are \(x=0, x=-1/2, x=3\) and y-intercept is -3. c) the function is positive for \(-1/2 < x < 3\), d) the function is negative for \(-∞ < x < -1/2\) and \(3 < x < ∞\), e) is in step 4.

Step 1: Find function with the same end behavior

The polynomial is \(f(x)=-x(2 x+1)(x-3)\) and can be rewritten as \(f(x)= -2x^3 + 5x^2 - 3x\). The leading term is \(-2x^3\), so a function with the same end behavior is \(y=-2x^2\).

Step 2: Find x- and y-intercept

Setting the function equal to zero will give the x-intercepts: \(-x(2 x+1)(x-3)=0\). The solutions are \(x=0, x=-1/2, x=3\). The y-intercept is found by substituting x=0 into the function, to get \(f(0) = -3\). So the y-intercept is -3.

Step 3: Determine where the function is positive or negative

The function is zero at \(x=0, x=-1/2, x=3\), these values split the x-axis into intervals. Testing a number from each interval in the function will indicate whether the function is positive or negative on that interval. The intervals are \(-∞ < x < -1/2\) (negative), \(-1/2 < x < 0\) (positive), \(0 < x < 3\) (positive), \(3 < x < ∞\) (negative).

Step 4: Sketch a graph using data

Based on this information, the sketch of the function should show the function opening downwards like \(y=-2x^2\), crossing the x-axis at \(x=0, x=-1/2, x=3\), crossing the y-axis at \(y=-3\), and being positive between \(x=-1/2\) and \(x=3\).

Key Concepts

End Behavior of Polynomials

The end behavior of a polynomial function describes how the function behaves as the input values (\( x \) values) become extremely large or extremely small. It's essentially about understanding what happens to the values of the polynomial as you go further out along the x-axis in both directions.

For the given polynomial function, \( f(x) = -x(2x+1)(x-3) \), rewriting it gives \( f(x) = -2x^3 + 5x^2 - 3x \). Here, the leading term is \( -2x^3 \), which determines the end behavior. In \(y = -2x^3\), the negative coefficient indicates that as \( x \) goes to positive infinity, \( f(x) \) goes to negative infinity, and as \( x \) goes to negative infinity, \( f(x) \) goes to positive infinity.

A simpler function with similar end behavior can be expressed as \( y = -2x^2 \). Even though it has a different degree, this quadratic polynomial with a negative leading coefficient also indicates that both ends of the graph will tend downwards, mimicking the original function's end behavior.

Intercepts of Polynomial Graphs

Intercepts of a polynomial graph are points where the graph crosses the axes. These are vital in understanding the behavior and placement of the graph on a coordinate plane.

The x-intercepts occur where the function equals zero, i.e., solving \( -x(2x+1)(x-3) = 0 \), resulting in \( x = 0 \), \( x = -\frac{1}{2} \), and \( x = 3 \). Each intercept reveals a point where the graph touches or intersects the x-axis.

The y-intercept is found by substituting \( x = 0 \) into the function. For \( f(x) = -2x^3 + 5x^2 - 3x \), evaluating \( f(0) \) results in \( y = -3 \). Thus, the y-intercept is at \( (0, -3) \).

These intercepts help plot the basic framework for sketching the polynomial graph and understanding where it crosses the axes.

Positive and Negative Intervals of Functions

Discovering where a function is positive or negative involves evaluating the sign of the function between intercepts. These evaluations can show where the graph lies above or below the x-axis.

For \( f(x) = -x(2x+1)(x-3) \), we set the x-intercepts at \( x = 0 \), \( x = -\frac{1}{2} \), and \( x = 3 \) to define intervals along the real number line:

• From \(-∞\) to \(-\frac{1}{2}\): By substituting any value in this interval into the function, we find it yields negative values.
• From \(-\frac{1}{2}\) to \(0\): Testing this interval with a chosen value gives positive results, indicating that the function is above the x-axis here.
• From \(0\) to \(3\): Any number used in this range likewise shows that the function remains positive.
• From \(3\) to \(+∞\): Checking this interval reveals negative function values, suggesting the graph drops below the x-axis.

The information derived from these intervals provides insights into the overall shape of the polynomial's graph, detailing where it rises and falls.