Problem 2 - Algebra 2

Question

Graph each polynomial function by making a table of values. $$ f(x)=x^{4}-7 x^{2}+x+5 $$

Step-by-Step Solution

Answer

Graph the function with the points: \((-3, 20), (-2, -9), (-1, -2), (0, 5), (1, 0), (2, -5), (3, 26)\).

Step 1: Create a Range of x-values

Select a range of x-values around which you want to observe the behavior of the polynomial. For a simple analysis, choose values such as \(-3, -2, -1, 0, 1, 2,\) and \(3\).

Step 2: Calculate the Corresponding f(x) Values

Substitute each x-value into the function to find the corresponding \(f(x)\) value. - For \(x=-3\), \(f(-3) = (-3)^4 - 7(-3)^2 + (-3) + 5 = 81 - 63 - 3 + 5 = 20\).- For \(x=-2\), \(f(-2) = (-2)^4 - 7(-2)^2 + (-2) + 5 = 16 - 28 - 2 + 5 = -9\).- For \(x=-1\), \(f(-1) = (-1)^4 - 7(-1)^2 + (-1) + 5 = 1 - 7 - 1 + 5 = -2\).- For \(x=0\), \(f(0) = 0^4 - 7(0)^2 + 0 + 5 = 5\).- For \(x=1\), \(f(1) = 1^4 - 7(1)^2 + 1 + 5 = 1 - 7 + 1 + 5 = 0\).- For \(x=2\), \(f(2) = 2^4 - 7(2)^2 + 2 + 5 = 16 - 28 + 2 + 5 = -5\).- For \(x=3\), \(f(3) = 3^4 - 7(3)^2 + 3 + 5 = 81 - 63 + 3 + 5 = 26\).

Step 3: Construct the Table of Values

Organize the x-values and corresponding \(f(x)\) values into a table to make it easier to plot.

Step 4: Plot the Points on a Graph

Using the table of values, plot the points on the coordinate plane. The points are: - \((-3, 20)\)- \((-2, -9)\)- \((-1, -2)\)- \((0, 5)\)- \((1, 0)\)- \((2, -5)\)- \((3, 26)\)

Step 5: Draw the Curve

Connect the plotted points with a smooth curve that best represents the shape of the polynomial function's graph. Since this is a degree 4 polynomial with a positive leading coefficient, the ends of the graph should rise on both sides.

Key Concepts

Polynomials

A polynomial is a mathematical expression consisting of variables and coefficients, which involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. In simpler terms, it's like putting building blocks together to form an expression. The general form of a polynomial is:

• \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_2 x^2 + a_1 x + a_0\)

where each \(a_i\) is a constant, and \(n\) is the degree of the polynomial. Polynomials are incredibly versatile and used in various areas of mathematics and science.
If you take our example function \(f(x) = x^4 - 7x^2 + x + 5\), it combines four terms, each a multiple of a power of \(x\). Notice how the exponents decrease from 4 down to 0, always taking non-negative values.

Graphing Techniques

Graphing a polynomial function can be a straightforward task when approached step by step. Start by choosing a range of \(x\)-values that you want to include in your graph. This selection should be wide enough to capture the behavior and shape of the polynomial. When plotting, utilize the following techniques:

• Plot a variety of points to accurately represent changes in direction and curvature.
• Observe symmetry or repetitive patterns which might help in making the graphing process quicker.
• Pay attention to intercepts, both \(x\) and \(y\), as these provide essential information about the polynomial.

Once the points are plotted, connect them smoothly to form the polynomial's curve. For degree 4 polynomials like ours, the graph should generally look like a wave, with two turning points, since it has a positive leading coefficient, the ends will rise.

Degree of Polynomials

The degree of a polynomial is one of its most important features. It reflects the highest power of the variable \(x\) in the expression. This degree can tell you a lot about the shape and behavior of the graph.

• A degree 4 polynomial will typically have 3 turning points, creating a complex wave shape.
• The degree also indicates the maximum number of roots or x-intercepts it may have.
• Moreover, the polynomial's end behavior (how the graph behaves as \(x\) approaches infinity) can often be determined from the leading term.

For our function \(f(x) = x^4 - 7x^2 + x + 5\), it's a fourth-degree polynomial, which means it typically has a smooth, continuous graph with four arms going up as \(x\) moves towards positive and negative infinity.

Table of Values

Creating a table of values is a crucial step in graphing polynomial functions, as it sets the stage for accurate plotting. Start by selecting a range of \(x\)-values, typically including positive, negative, and zero, to give a balanced view of the function's behavior.

• Plug each \(x\)-value into the polynomial function to compute the corresponding \(f(x)\) value.
• Record these results in a table, with one column for \(x\)-values and another for \(f(x)\)-values.
• Organizing data in this way makes it much easier to spot trends and prepare for graphing.

In our example, by substituting values like \(-3\), \(-2\), \(-1\), \(0\), \(1\), \(2\), and \(3\), we obtain coordinates such as \((-3, 20)\) and \((1, 0)\). These points help guide the sketch of \(f(x) = x^4 - 7x^2 + x + 5\) on a coordinate plane, ensuring it molds to the correct shape.