Problem 16 - Algebra and Trigonometry

Question

\(15-26\) Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=(x-1)(x+1)(x-2) $$

Step-by-Step Solution

Answer

The graph is a cubic curve passing through \((-1,0)\), \((1,0)\), and \((2,0)\) with a y-intercept at \((0, 2)\), rising from \(-\infty\) to \(+\infty\).

Step 1: Understand the Polynomial

The given polynomial is \(P(x) = (x-1)(x+1)(x-2)\). This is a cubic polynomial because it can be expanded to have the standard form \(P(x) = x^3 - 2x^2 - x + 2\). Since it's a cubic polynomial, its general shape will involve one or two turns and an end behavior that falls to different extremes as \(x\) approaches \(\pm \infty\).

Step 2: Find the Roots

The roots of the polynomial are \(x=1\), \(x=-1\), and \(x=2\). These are found by solving each factor set equal to zero: \((x-1)=0\), \((x+1)=0\), and \((x-2)=0\). Each root is an x-intercept of the graph.

Step 3: Determine the End Behavior

Since the leading term of \(P(x)\) is \(x^3\), and this term dominates as \(|x|\) becomes large, the end behavior is that as \(x\to +\infty), \; P(x) \to +\infty\), and as \(x \to -\infty, \; P(x) \to -\infty\).

Step 4: Find the y-Intercept

The y-intercept occurs where \(x=0\). Substitute 0 into the function: \(P(0) = (0-1)(0+1)(0-2) = -1(-2) = 2\). Therefore, the y-intercept is \((0, 2)\).

Step 5: Sketch the Graph

Plot the x-intercepts \(x=-1\), \(x=1\), and \(x=2\), and the y-intercept \((0, 2)\) on a coordinate plane. Using the end behavior, draw the curve starting from \(-\infty\) passing through the intercepts and following the pattern of the polynomial to \(+\infty\). The graph should curve up through \(x=-1\), drop to \(y=2\) at \(x=0\), dip down through \(x=1\), and rise again through \(x=2\).

Key Concepts

Cubic Polynomial

A cubic polynomial is a type of polynomial that has the largest exponent, or degree, of three. This characteristic is reflected in terms of the form \( ax^3 + bx^2 + cx + d \). For our given polynomial, \( P(x) = (x-1)(x+1)(x-2) \), when expanded it becomes \( x^3 - 2x^2 - x + 2 \).
This confirms it's a cubic polynomial. The general shape of a cubic polynomial's graph involves subtle twists and turns primarily due to the degree of the polynomial. Such polynomials usually change direction around their local minima and maxima.
Because of their cubic nature, these polynomials usually have a more complex curve compared to lower-degree polynomials like linear or quadratic ones.

Roots of Polynomial

The roots of a polynomial are the values of \(x\) for which the polynomial evaluates to zero. They are also known as solutions of the equation or x-intercepts of the graph. For the polynomial \( P(x) = (x-1)(x+1)(x-2) \), the roots are quite straightforward to find. These are obtained by setting each factor equal to zero:

• \(x-1=0\) which means \(x=1\)
• \(x+1=0\) which means \(x=-1\)
• \(x-2=0\) which means \(x=2\)

These roots indicate locations where the graph of the polynomial will intersect the x-axis. These intersection points give us a guide on where the graph will pass, helping form the overall shape.

End Behavior

End behavior refers to how the graph of a polynomial behaves as \( x \) approaches positive or negative infinity. It is primarily determined by the leading term of the polynomial, which in our polynomial is \( x^3 \).
Leading terms have the greatest impact due to their higher degree, especially when \( |x| \) becomes large.For our cubic polynomial, the leading term is positive \( x^3 \), which means that:

• As \( x \to +\infty \), \( P(x) \to +\infty \)
• As \( x \to -\infty \), \( P(x) \to -\infty \)

This behavior ensures that, over long stretches, the graph will rise towards positive infinity to the right and fall towards negative infinity to the left. Understanding this aspect helps sketch the gross movements of the polynomial's graph.

Intercepts of Graph

Intercepts are critical points where the graph of the polynomial crosses the axes. They provide anchoring points that define the interaction of the polynomial with the coordinate plane.

• X-intercepts: These are the points where the graph crosses the x-axis. For \( P(x) = (x-1)(x+1)(x-2) \), the x-intercepts are \(x=-1\), \(x=1\), and \(x=2\).
• Y-intercept: This is the point where the graph crosses the y-axis. It is found by evaluating the polynomial at \(x=0\). In our case, substituting \(x = 0\) gives \(P(0) = 2\), so the y-intercept is \((0, 2)\).

Understanding these intercepts allows for a more accurate and meaningful sketch of the cubic polynomial graph, providing visible references and confirmation of the polynomial's properties when graphed.