Problem 11 - Algebra 2

Question

For Exercises \(11-18,\) complete each of the following. a. Graph each function by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=-x^{3}-4 x^{2} $$

Step-by-Step Solution

Answer

The function's zero is at \( x = 0 \). Relative extrema can be located at critical points \( x = 0 \) and \( x = -\frac{8}{3} \).

Step 1: Creating a Table of Values

To graph the function, we first make a table of values for \( f(x) = -x^3 - 4x^2 \). Choose values of \( x \) around the real zeros and points of interest. Calculate \( f(x) \) for each value:| \( x \) | \( f(x) \) ||---|---|| -3 | \( -(-3)^3 - 4(-3)^2 = -27 - 36 = -63 \) || -2 | \( -(-2)^3 - 4(-2)^2 = -8 - 16 = -24 \) || -1 | \( -(-1)^3 - 4(-1)^2 = -1 - 4 = -5 \) || 0 | \( -(0)^3 - 4(0)^2 = 0 \) || 1 | \( -(1)^3 - 4(1)^2 = -1 - 4 = -5 \) |

Step 2: Plotting the Graph

Using the table of values, plot each point on a coordinate grid. Connect the points with a smooth curve to represent the function \( f(x) = -x^3 - 4x^2 \). The graph should extend from left to right, showing the curve trending downwards due to the negative coefficients, with no symmetry.

Step 3: Identifying Real Zeros

Real zeros are points where the graph crosses the x-axis. From the table, we see that \( f(0) = 0 \), indicating a zero at \( x = 0 \). As \( f(x) \) changes from positive to negative or vice versa around zero, check for crossings between calculated points like -2 to 1.

Step 4: Finding Consecutive Integers for Real Zeros

Look for sign changes in \( f(x) \) in the table values. The function crosses the x-axis between integers without another visible zero, indicating the zero is between limited points. The zero is at \( x = 0 \), hence no need to further estimate.

Step 5: Estimating Relative Extrema

To find relative maxima or minima, differentiate \( f(x) \):\[ f'(x) = -3x^2 - 8x \]. Setting \( f'(x) = 0 \) gives:\[ -3x^2 - 8x = 0 \].Factor out \( x \) to solve:\[ x(-3x - 8) = 0 \].Thus, \( x = 0 \) or \( x = -\frac{8}{3} \). These are critical points; check nearby values to confirm slope behavior and extremum.

Key Concepts

Real Zeros

Real zeros are where the graph of a polynomial intersects the x-axis. For the function \( f(x) = -x^3 - 4x^2 \), these points can be determined by solving the equation \( f(x) = 0 \). This involves finding the values of \( x \) where the function equals zero.

From the step by step solution, we identified that one of the real zeros is at \( x = 0 \). This was found directly by calculating \( f(0) \), which equals zero. The correct identification of real zeros is essential because they show where the graph shifts above or below the x-axis.

An important aspect when finding zeros is checking for sign changes around your calculated points. In our function, looking between values like \( x = -2 \) and \( x = 1 \) reveals potential crossings. This indicates that between these intervals the graph experiences a change in the y-axis direction, thereby confirming these are close to where the real zeros lie.

Relative Extrema

Relative extrema refer to the points on a graph where the function reaches a local maximum or minimum. These are points where the graph changes direction from increasing to decreasing or vice versa.

To find these points for the function \( f(x) = -x^3 - 4x^2 \), you need to differentiate the function. By solving \( f'(x) = 0 \), you determine the derivative's critical points, \( x = 0 \) and \( x = -\frac{8}{3} \).

These critical points are then tested by checking nearby x-values to understand the behavior of the function around them. For instance, if the derivative changes from negative to positive at these points, then it indicates a relative minimum, whereas a change from positive to negative suggests a relative maximum. This detailed analysis helps to understand where the curve bends and reaches high or low points.

Derivative

The derivative of a function gives the rate of change of the function’s output with respect to its input. It is a crucial tool in determining the function’s slope at any given point, which in turn can tell us about the behavior of the graph.

For the polynomial \( f(x) = -x^3 - 4x^2 \), the derivative is calculated as \( f'(x) = -3x^2 - 8x \). This expression allows us to find where the slope of the function is zero, indicating potential points of relative extrema.

Once \( f'(x) = 0 \) is solved, i.e. \[ -3x^2 - 8x = 0 \], it illustrates where changes in the graph's direction occur, pinpointing critical points. Thus, derivatives are instrumental in graphing functions accurately and understanding their underlying dynamics.

Table of Values

Creating a table of values is a vital step before graphing any function. This table includes various \( x \) inputs alongside their corresponding \( f(x) \) outputs. By choosing a range of \( x \)-values, especially those surrounding important features like zeros and extrema, we acquire a clear picture of the function's behavior.

In our example \( f(x) = -x^3 - 4x^2 \), the table of values comprised values like:

• \( x = -3 \), \( f(x) = -63 \)
• \( x = -2 \), \( f(x) = -24 \)
• \( x = -1 \), \( f(x) = -5 \)
• \( x = 0 \), \( f(x) = 0 \)
• \( x = 1 \), \( f(x) = -5 \)

By plotting these points on a graph, and connecting them with a smooth curve, the graph of the polynomial is unveiled. A comprehensive table of values is the foundation for sketching the most accurate representation of polynomial functions, facilitating both visual analysis and finding critical points.