Problem 13 - Algebra for College Students

Question

For Problems \(1-22\), graph each of the polynomial functions. $$ f(x)=x(x+2)(2-x) $$

Step-by-Step Solution

Answer

Roots are -2, 0, 2; y-intercept is 0; end behavior is affected by -x^3. Graph shows these characteristics.

Step 1: Expand the Polynomial

To graph the polynomial function, we first need to express it in standard form. Start by expanding the expression \(f(x) = x(x + 2)(2 - x)\). First, simplify the terms inside. Multiply \((x + 2)(2 - x)\) to get:\[(x + 2)(2 - x) = 2x - x^2 + 4 - 2x = -x^2 + 4\].Therefore, the function becomes \(f(x) = x(-x^2 + 4)\). Expanding this gives us:\[f(x) = -x^3 + 4x\].This is now in its polynomial form.

Step 2: Determine the Roots

The roots of the polynomial (the x-intercepts of the graph) can be found from the factored form \(x(x+2)(2-x)\). These are the solutions to \(x(x+2)(2-x)=0\).The roots are: - \(x = 0\)- \(x = -2\)- \(x = 2\)These roots mean the polynomial crosses the x-axis at these points.

Step 3: Identify the End Behavior

Since the leading term in the expanded version \(f(x) = -x^3 + 4x\) is \(-x^3\), which is of odd degree and has a negative leading coefficient, the end behavior of the polynomial will be:- As \(x \to \infty\), \(f(x) \to -\infty\)- As \(x \to -\infty\), \(f(x) \to \infty\)This affects the graph's direction at the far ends.

Step 4: Compute the y-Intercept

The y-intercept is where the graph crosses the y-axis. This occurs where \(x=0\). Evaluating the polynomial we expanded, we have:\[f(0) = -(0)^3 + 4(0) = 0\]Thus, the y-intercept is also (0,0), which confirms our root at this point.

Step 5: Sketch the Graph

Now, use the information collected to sketch the graph:- Plot the roots \(x = -2\), \(x = 0\), and \(x = 2\) on the x-axis.- Add the y-intercept at \((0, 0)\).- From the end behavior analysis, draw the left branch of the graph up to \(x = -2\), it passes through the x-axis and bends back down through the root \(x = 0\), continues upwards through \(x = 2\), and finally heads downward as \(x\) increases.- Ensure the curve correctly reflects the cubic shape expected based on the polynomial \(-x^3 + 4x\). This gives you an accurate visual representation of the function.

Key Concepts

Polynomial Expansion

Polynomial expansion involves rewriting a polynomial function in its standard form. This process is crucial to understand the behavior and detailed features of the polynomial. In our given problem, the original function is expressed as a product of factors:

• Start by tackling the expression \(f(x) = x(x + 2)(2 - x)\).
• Expand \((x + 2)(2 - x)\) first, resulting in:\(-x^2 + 4\).
• Now multiply this result by \(x\), giving the expanded polynomial:\(f(x) = -x^3 + 4x\).

Expanding polynomials helps in identifying terms that contribute to the shape of the graph and provides a clear representation which is useful for graphing purposes.

Roots of Polynomial

The roots of a polynomial, also known as the x-intercepts, are the values for which the polynomial equals zero. These points are vital for understanding where the graph of the polynomial intersects the x-axis. Given the factored form of the polynomial:

• \(x(x+2)(2-x)\), we find the roots by setting each factor equal to zero:
• \(x = 0\)
• \(x + 2 = 0\) leads to \(x = -2\)
• \(2 - x = 0\) gives \(x = 2\)

These root calculations confirm that the graph will cross the x-axis at \(-2, 0,\) and \(2\). Recognizing the roots serves as a guide for where your graph needs to intersect the axis.

End Behavior of Polynomials

The end behavior of a polynomial function describes how the function behaves as the values of \(x\) increase positively or negatively towards infinity. This is mainly determined by the leading term of the polynomial. In our expanded polynomial:

• \(f(x) = -x^3 + 4x\), the leading term is \(-x^3\).
• The degree is 3 (odd) and the coefficient is negative.
• This means as \(x \to \infty\), \(f(x) \to -\infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).

Understanding these trends helps predict how the polynomial graph will extend and in which direction the arms of the graph will point.

Y-Intercept

The y-intercept of a polynomial graph is the point at which the graph crosses the y-axis. This occurs where \(x = 0\). To find the y-intercept, substitute \(x = 0\) into the expanded polynomial \(f(x) = -x^3 + 4x\):

• \(f(0) = -(0)^3 + 4(0) = 0\).

This calculation confirms that the y-intercept is at \((0,0)\). This spot is crucial when sketching the graph, providing another anchor point alongside the roots. Hence, confirming and plotting the y-intercept helps solidify the overall shape and positioning of the polynomial graph on the coordinate plane.