Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Answer true or false to each statement. Then support your answer by graphing. The graph of \(f(x)=x^{3}-3 x^{2}+3 x-1=(x-1)^{3}\) has exactly one \(x\) -intercept.

Step-by-Step Solution

Answer

True; the graph has exactly one x-intercept at (1, 0).

Step 1: Understand the Function

The given function is \[f(x) = x^3 - 3x^2 + 3x - 1\].We can also express it as:\[f(x) = (x-1)^3\].This indicates it is a cubic function that has been transformed.

Step 2: Identify the X-Intercept

To find the x-intercept of a function, set \(f(x) = 0\) and solve for \(x\). Here, we solve:\[(x-1)^3 = 0\].This gives:\[x-1 = 0\] or \[x = 1\].Hence, the x-intercept is at \((1, 0)\).

Step 3: Analyze the Multiplicity

The function \((x-1)^3 \) indicates the root \(x = 1\) has a multiplicity of 3. A root with odd multiplicity crosses the x-axis. Therefore, there is only one x-intercept at \(x = 1\).

Step 4: Graph the Function

Sketch the graph of \(f(x) = (x-1)^3\). Since this is a cubic function with a single root at \(x=1\), the graph will pass through the x-axis at \(x=1\) and appear as an S-shape since it is an odd-degree polynomial with positive leading coefficient.

Key Concepts

Polynomial Graphs

Polynomial graphs are visual representations of polynomial functions. These functions can vary in degree, where the degree is determined by the highest power of the variable present in the function. For example, a cubic function is a polynomial of degree three, like our function \( f(x) = (x-1)^3 \). These graphs help illustrate how polynomial functions behave under different conditions and are crucial for understanding the function's roots, intercepts, and end behavior. The overall shape of the graph depends on the leading coefficient and the degree of the polynomial, dictating whether the graph will have turning points, cross the axes, or continue indefinitely in a particular direction.Key features of polynomial graphs:

• The degree of the polynomial determines the maximum number of x-axis crossings (also known as real roots) and turning points, minus one.
• The leading coefficient can indicate whether the ends of the graph extend upwards or downwards.
• Cubic polynomials often create an "S"-shaped curve due to their odd degree and can display one or more changes in direction.

Studying polynomial graphs helps in visualizing the solutions to polynomial equations and understanding how changes in equations affect the graph's shape.

X-Intercepts

X-intercepts are points where the graph of a function crosses the x-axis. At these points, the value of the output or y-coordinate of the function is zero, i.e., \( f(x) = 0 \). In our exercise, we determined the x-intercept by solving the equation\( (x-1)^3 = 0 \).To find x-intercepts:

• Set the polynomial function equal to zero: \( f(x) = 0 \).
• Solve for the variable \( x \), which will give you the x-coordinates of the intercepts.

For the function \( f(x) = (x-1)^3 \), this process provides a solution of \( x = 1 \), meaning the graph touches or crosses the x-axis at the point (1,0).Graphically, x-intercepts are significant as they represent the roots of the equation. An x-intercept shows us exactly where the function changes sign. Understanding x-intercepts allows one to predict and analyze the behavior of the polynomial across its domain.

Multiplicity of Roots

Multiplicity of roots in polynomial equations relates to how many times a particular solution, or root, is repeated. To determine multiplicity, you examine the power to which the factor corresponding to a root is raised.In the function \( f(x) = (x-1)^3 \), the root \( x = 1 \) appears with a multiplicity of 3. This means that \( (x-1) \) is repeated thrice in the factorization.The impact of multiplicity on the graph:

• If the multiplicity is odd, like in this case, the graph will cross the x-axis at that point.
• If it is even, the graph will only touch and not cross the x-axis, appearing as if it "bounces" off.
• Higher multiplicities cause the graph to flatten out near the root.

Understanding multiplicity helps in predicting the appearance of the graph at each root, providing insight into more intricate features of the polynomial without having to graph it explicitly.