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Chapter 2: Problem 62

Graphical Analysis In Exercises \(59-64,\) use a graphing utility to graph the quadratic function. Find the \(x\) -intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation when \(f(x)=0 .\) $$ f(x)=x^{2}-8 x-20 $$

Short Answer

Expert verified
The x-intercepts of the graph of the function \(f(x) = x^{2} - 8x - 20\) are the solutions of the equation when \(f(x) = 0\). Identifying these intercepts graphically and solving the equation \(x^{2} - 8x - 20 = 0\) algebraically gives the same results, thus validating the solutions.

Step by step solution

01

Graph the function

First of all, use a graphing utility to sketch the graph of the quadratic equation \(f(x) = x^{2} - 8x - 20\). Observe the graph to determine where it intersects the x-axis, these points are known as x-intercepts.
02

Identify the x-intercepts from the graph

The x-intercepts of the graph represent the values of x when \(f(x) = 0\). In other words, they are the solutions of the equation \(x^{2} - 8x - 20 = 0\). Identify these points on the graph.
03

Solve the equation algebraically

While simultaneous comparison with the graph, solve the equation algebraically by first putting the function equal to zero, \((x^{2} - 8x - 20 = 0)\), then factorizing it or using the quadratic formula to find the roots \(x=(8\pm \sqrt{(8^2-4(1)(-20))})/(2*1)\). This will give two solutions.
04

Comparison

Finally, compare the x-intercepts from the graphical solution with the solutions from the algebraic method. They should coincide with each other, validating the correctness of both methods.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

X-Intercepts of a Parabola
Understanding the x-intercepts of a parabola is a fundamental aspect of graphing quadratic functions. These points where the parabola crosses or touches the x-axis are crucial in determining the shape and position of the graph. Here's how you can find the x-intercepts:

To determine the x-intercepts algebraically, you set the quadratic function to zero, essentially solving the equation for when the output, or f(x), is equal to zero. A quadratic function in the form of f(x) = ax^2 + bx + c can have two, one, or no real x-intercepts depending on the discriminant, which is given by b^2 - 4ac. If the discriminant is positive, there will be two real intercepts, if it's zero there will be one, and if it's negative, there will be no real x-intercepts, indicating the parabola does not touch the x-axis.

Example:

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Most popular questions from this chapter

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