Problem 53 - Precalculus: Functions and Graphs

Question

Graph \(f,\) and estimate its zeros. $$f(x)=x^{3}-3 x+1$$

Step-by-Step Solution

Answer

The zeros of the function are approximately at \(x \approx -1.88, 0.53, 1.35\).

Step 1: Identify the Function

We are given the function \(f(x) = x^3 - 3x + 1\). Our task is to graph this function and estimate its zeros.

Step 2: Find Critical Points

To help in graphing, find the derivative \(f'(x) = 3x^2 - 3\), and set it to zero to find the critical points: \[ 3x^2 - 3 = 0 \]\[ x^2 = 1 \]\[ x = \pm 1 \] These points may be where the function changes its increasing or decreasing nature.

Step 3: Evaluate the Function at Critical Points

Compute the function values at the critical points to help sketch the graph:\[ f(1) = 1^3 - 3(1) + 1 = -1 \]\[ f(-1) = (-1)^3 - 3(-1) + 1 = 3 \] This gives us the points (1, -1) and (-1, 3) on the graph.

Step 4: Sketch the Graph

Plot the points identified: the y-intercept \((0,1)\), and the critical points \((1, -1)\) and \((-1, 3)\). Since the function is a cubic polynomial and since the leading term is \(x^3\), the graph will continue as \(x o -fty\) \(f(x) o -fty\) and \(x o fty\) \(f(x) o fty\). Use this information to sketch the curve.

Step 5: Estimate Zeros from the Graph

Once the graph is sketched, find where the curve intersects the x-axis. These are the estimated zeros of the function.

Key Concepts

Critical Points

In the context of graphing cubic functions, critical points are crucial as they potentially mark where the function's graph changes direction. To find these points, one must compute the derivative of the function.
Critical points occur where the derivative equals zero or is undefined. For the cubic function \(f(x) = x^3 - 3x + 1\), the derivative is determined as \(f'(x) = 3x^2 - 3\).
By setting the derivative to zero, \(3x^2 - 3 = 0\), we solve for \(x\) and find the critical points at \(x = \pm 1\). These points are helpful in determining where the graph may change from increasing to decreasing, or vice versa.

• Identifying critical points ensures a more accurate sketch of the graph.
• They indicate possible maximum or minimum points of the function.

Function Derivatives

Understanding function derivatives is fundamental in analyzing and graphing polynomial functions like the one given here.
The derivative of a function describes the rate at which the function's value changes with respect to changes in its input value \(x\). The derivative \(f'(x) = 3x^2 - 3\) gives us critical information about the behavior of the original function \(f(x) = x^3 - 3x + 1\).For polynomial functions, the derivative helps identify critical points by revealing changes in direction of the function:

• The derivative being zero at \(x = 1\) and \(x = -1\) pinpoints these as places where the slope of the tangent to the curve is horizontal.
• This can suggest local maxima, minima, or even inflection points.

Zeros of a Function

The zeros of a function are vital in understanding the graphical representation of polynomial functions. These are the points where the graph of the function crosses or touches the x-axis.
For our function \(f(x) = x^3 - 3x + 1\), identifying the zeros involves examining the x-coordinates where \(f(x) = 0\).Zeros can be estimated from the sketched graph:

• Look where the curve intersects the x-axis. These intersections are the function's zeros.
• Cubic functions generally have up to three zeros since their highest degree is three.

Understanding the function's zeros lets us deduce important insights about the roots and solutions of the equations formed by these polynomial functions.

Polynomial Functions

Polynomial functions, such as \(f(x) = x^3 - 3x + 1\), are expressions involving powers of \(x\). They are influential because they can be used to model various real-world phenomena and are immensely helpful in algebra and calculus.
Each polynomial has a degree that dictates its shape and nature:

• The polynomial's degree indicates the maximum number of turning points and intersections with the x-axis (zeros).
• Cubic polynomials, which are degree 3, may have a characteristic "S" shape and can change direction one or two times.

Interpreting and graphing polynomial functions require identifying key points like the y-intercept, critical points, and zeros to sketch an accurate graph.