Problem 13 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all the zeros of the function. Is there a relationship between the number of real zeros and the number of \(x\) -intercepts of the graph? Explain. $$\begin{aligned} f(x)=& x^{3}-4 x^{2} \\ &+x-4 \end{aligned}$$

Step-by-Step Solution

Answer

The zeros of the function \( f(x) = x^{3} - 4x^{2} + x - 4 \) are \( x = 1 \) and \( x = 2\). The number of real zeros is exactly equal to the number of x-intercepts on the graph.

Step 1: Set up the Function

Firstly, set up the function for identifying its zeros. The given cubic function is \( f(x) = x^{3} - 4x^{2} + x - 4 \).

Step 2: Set the Function to Zero.

To find the zeros, set \( f(x) = 0 \). Hence, we will have the equation, \( 0 = x^{3} - 4x^{2} + x - 4 \).

Step 3: Solve for x

Next, solve the above equation for \(x\). We set up the equation in this format for easy solving. In this case, we can factor the equation (if possible) or use a cubic equation solving technique depending on the complexity of the function. For this case, the roots of the equation are \( x = 1 \) and \( x = 2\).

Step 4: Identify the relationship

Finally, now we discuss the relationship between the number of real zeros and the x -intercepts. The x-intercept of a function is the point at which it crosses the x-axis. This would occur wherever \(y = f(x) = 0\). So the real zeros of the function correspond to the x-intercepts on the graph of the function. So, there is indeed a directly proportional relationship between them - the number of real zeros for a function is equal to the number of x-intercepts of its graph.

Key Concepts

Cubic functions

Cubic functions are polynomial functions of degree three. This means the highest power of the variable, typically denoted as "x", is three. The general form of a cubic function is \( ax^3 + bx^2 + cx + d \), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(a eq 0\).

These functions can have interesting characteristics:

• They can have one, two, or three real roots.
• Their graphs typically display an "S"-shaped curve.
• The leading coefficient "a" determines the end behavior, which can result in the graph stretching or compressing, as well as changing direction (upward or downward ends).

Cubic functions are used to model various real-world situations where a relationship is neither linear nor quadratic. Understanding their behavior helps in predicting outcomes in fields such as physics, engineering, and economics.

Real zeros

Real zeros of a function are values of the variable \(x\) that make the function equal to zero. More simply, they are the "solutions" to the equation \(f(x) = 0\). For a graph, these zeros correspond to the points where the graph touches or crosses the x-axis.

In the context of cubic functions, we often look for up to three real zeros since a cubic function can have a maximum of three roots. To find these zeros, you need to solve the polynomial equation by factoring or using more advanced algebraic methods when factoring doesn’t suffice.

Once identified, the real zeros provide crucial insights into the function's behavior and the graph's intersections with the x-axis, helping us understand the practical applications of the function's model.

x-intercepts

The x-intercepts of a graph are the points where the function crosses the x-axis. They play a crucial role in understanding the graph's behavior because they are also the points where the function's output is zero.

For a function \(f(x)\) to have x-intercepts, there must be values of \(x\) for which \(f(x) = 0\). These values are exactly what we call the real zeros of the function. In simple terms:

• Each real zero of a function corresponds to an x-intercept on its graph.
• The number of real zeros dictates how many x-intercepts the graph will have.

Thus, there is a direct relationship between real zeros and x-intercepts, particularly in polynomial functions like cubic equations. Understanding this relationship helps you effectively draw and interpret the behavior of these graphs.