Question

Sketch a graph of a polynomial function \(f\) having the given characteristics. \- The graph of \(f\) has \(x\)-intercepts at \(x=-3, x=1\), and \(x=5\). \- \(f\) has a local maximum value when \(x=1\). \- \(f\) has a local minimum value when \(x=-1\) and when \(x=3\).

Short answer

Function sketch will start falling from the left, rise starting from \(-3\) (local minimum), reach a peak at \(x=1\) (local maximum), drop back down at \(x=3\) (local minimum), and go up indefinitely from \(x=5\), resembling a W. The actual function could be given as \(f(x) = -a(x+3)^2(x-1)(x-5)\), where a > 0.

Step-by-step solution

Express function with zeros

First, setup the basic equation of the polynomial with given zeros: \(f(x) = a(x + 3)(x - 1)(x - 5)\), where \(a\) is a constant.

Identify Local Extrema

From the given local maxima and minima, we can see that the function decreases from \(-\infty\) to -1, increases to 1, decreases to 3, and increases again to \(+\infty\) (i.e. the function flips its monotonicity three times). This suggests that our function might be of fourth-degree.

Use additional properties

We know it's a fourth-degree polynomial and we want it to have a negative leading coefficient (since it falls to the left and rises to the right), so our polynomial could be \(f(x) = -a(x+3)^2(x-1)(x-5)\). Notice that the (x+3) term is squared. This is because local maxima and minima occur when the derivative of the polynomial equals zero, and squaring the (x+3) term will ensure that there is a local minimum at x = -3.

Sketch the graph

We do not need to determine the actual value of \(a\), we only care about its sign. Plot all of these points and behaviors onto one graph, being sure to pay attention to end behavior and the given intercepts and extrema. This will give us visual of what our function looks like.

Key concepts

x-intercepts

In mathematical terms, an "x-intercept" is the point where a graph of a function crosses the x-axis. At these points, the value of the function is zero. In the context of polynomial functions, the x-intercepts are directly related to the roots or zeros of the polynomial.
For example, if you have a polynomial function such as \(f(x) = a(x+3)(x-1)(x-5)\), the x-intercepts can be found by setting the function equal to zero:

• \(f(x) = 0 \Rightarrow (x+3)(x-1)(x-5) = 0\)
• This directly gives us the intercepts at \(x = -3, 1, \) and \(5\).

X-intercepts are crucial in sketching the graph of a polynomial function because they provide key points where the graph will cross the x-axis. This can help outline the behavior of the graph across the different quadrants.

local extrema

Local extrema refer to points on a graph where the function reaches local maximum or minimum values. These points are important because they indicate a change in the direction of the function. When sketching the graph of a polynomial function, identifying these points helps to accurately depict the graph's shape.
For the function in the exercise, \(f\) has a local maximum at \(x=1\), and local minima at \(x=-1\) and \(x=3\). This means:

• The graph of the function will reach a peak (local max) at \(x=1\), and a valley (local minima) at \(x=-1\) and again at \(x=3\).

The presence of local extrema suggests specific dynamics in the function's behavior, such as intervals of increase or decrease. These intervals can determine how the graph of the polynomial transitions at different sections of the x-axis, creating a curve that reflects these ups and downs.

fourth-degree polynomial

A fourth-degree polynomial is distinguished by the highest power of the variable, which is four. This means the general form of such a polynomial is given by \(ax^4 + bx^3 + cx^2 + dx + e\). In this exercise, the function is determined to be of fourth-degree based on the behavior of the local maxima and minima.
The function's behavior changes direction three times between the given intervals, suggesting it is of higher degree. Specifically, a fourth-degree polynomial can have up to three local maxima or minima, which is consistent with the exercise stating a max at \(x=1\) and mins at \(x=-1\) and \(x=3\).
Higher-degree polynomials like fourth-degree ones can model more complex curves than lower degree polynomials, providing a flexible tool for representing real-world data with multiple turns.

leading coefficient

The leading coefficient is the coefficient in front of the highest degree term of a polynomial. It plays a crucial role in determining the end behavior of the polynomial function's graph. Specifically, for polynomial functions, the sign of the leading coefficient reveals how the graph behaves at the ends of the x-axis.
In our exercise, the polynomial is described to have a negative leading coefficient, indicating that as \(x\to -\infty\), the function falls (goes down), and as \(x\to \infty\), the function rises (goes up).
The polynomial form \(f(x) = -a(x+3)^2(x-1)(x-5)\) shows a negative sign for the leading coefficient, reinforcing the specified end behavior condition. This coefficient is key to correctly sketching the polynomial's graph, ensuring that it represents the appropriate dynamic by tilting the graph downwards on one side and upwards on the other.