Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Algebraic and Graphical Approaches In Exercises \(31-46\), find all real zeros of the function algebraically. Then use a graphing utility to confirm your results. $$f(x)=x^{2}-25$$

Step-by-Step Solution

Answer

The real zeros of the function \(f(x)=x^{2}-25\) are \(x=5\) and \(x=-5\).

Step 1: Solve the Equation Algebraically

The function \(f(x)=x^{2}-25\) is given. To find the zeros of the function, we set the function equal to zero and solve for \(x\). That gives us the quadratic equation \(x^{2}-25=0\). We start by adding 25 to both sides of the equation to isolate \(x^{2}\) on one side - this gives us \(x^{2}=25\). The square root of 25 is \(\pm5\). Hence, the equation \(x^{2}=25\) gives two solutions: \(x=5\) and \(x=-5\). These are the zeros of the function.

Step 2: Confirm the Solutions Graphically

Now we confirm our solutions using a graphing utility tool. To do this, graph the quadratic function \(y=x^{2}-25\), and identify key features such as intercepts and turning points. The function should intersect the x-axis at \(x=5\) and \(x=-5\), which confirms our algebraic solutions. This is because the x-intercepts represent the values of \(x\) for which the function equals zero.

Key Concepts

Real Zeros

Finding the real zeros of a quadratic equation is an essential part of understanding the behavior of the function. "Zeros" refer to the values of \( x \) for which the function \( f(x) \) equals zero. Real zeros are where the graph of the function crosses the x-axis in the coordinate plane. These are the solutions to the equation when set to zero.
To find the real zeros of the quadratic equation \( f(x) = x^2 - 25 \), start by setting the equation equal to zero, resulting in \( x^2 - 25 = 0 \). From here, add 25 to both sides to obtain \( x^2 = 25 \). The next step involves taking the square root of both sides, which gives two potential solutions: \( x = 5 \) and \( x = -5 \). These values are the real zeros of the function. Real zeros are important because they give us specific solutions where the function has no value, acting as key points in the graph's architecture.

Graphing Utility

Graphing utilities are powerful tools used to visualize functions, and gain insight into their characteristics. Most commonly, graphing utilities take the form of software applications or graphing calculators. These tools allow users to input a function, like \( y = x^2 - 25 \), and plot its graph on a coordinate plane.
In this exercise, after solving the equation algebraically, a graphing utility can confirm the zeros we found, \( x = 5 \) and \( x = -5 \). By graphing \( y = x^2 - 25 \) on a graphing utility, you will observe that the curve intersects the x-axis precisely at these points. The x-intercepts of the graph not only verify the zeros but also illustrate key aspects like the vertex, shape, and direction of the parabola.
Graphing utilities enhance understanding by letting you see a function's behavior and confirming algebraic calculations visually.

Algebraic Solution

Algebraic solutions provide a method to solve equations using fundamental algebra operations. This involves manipulating the equation to isolate the variable and results in precise numerical solutions.
For the function \( f(x) = x^2 - 25 \), an algebraic approach begins with the step of equating the function to zero, forming the equation \( x^2 - 25 = 0 \). By simplifying this equation through addition, subtraction, multiplication, or division, we seek to find the value of \( x \) that satisfies the equation. Here, adding 25 to both sides simplifies the equation to \( x^2 = 25 \). Taking the square root yields \( x = 5 \) and \( x = -5 \).
These steps solve the quadratic equation logically and clearly, offering a direct pathway to the solutions. As solutions coincide with the function's zeros and points of intersection on its graph, algebra serves as the foundational mechanism for uncovering these important features.