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Chapter 2: Problem 62

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 are 10 and -2. These are the solutions to the corresponding quadratic equation, thereby confirming their correctness.

Step by step solution

01

Graph the quadratic function

In a suitable graphing utility, input the quadratic equation \(f(x)=x^{2}-8x-20\). The graph will be a parabola.
02

Finding the x-intercepts

The x-intercepts of a function are the points where the function touches or crosses the x-axis. On the graph, these points can be found where the graph intersects the x-axis.
03

Solving the quadratic equation

Solving the equation \(f(x)=0\) gives \(x^2 - 8x -20 = 0\). Factoring the equation gives \((x-10)(x+2) = 0\). The solutions to this equation \(x = 10, -2\) are the x-intercepts of the graph of the quadratic equation.
04

Comparing solutions

The solutions from the quadratic equation and the x-intercepts from the graph of the function should be the same. Thus, we can see that they do match.

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Most popular questions from this chapter

The maximum safe load uniformly distributed over a one-foot section of a two- inch-wide wooden beam can be approximated by the model $$\text { Load }=168.5 d^{2}-472.1$$ where \(d\) is the depth of the beam. (a) Evaluate the model for \(d=4, d=6, d=8, d=10\) and \(d=12 .\) Use the results to create a bar graph. (b) Determine the minimum depth of the beam that will safely support a load of 2000 pounds.

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{4}+10 x^{2}+9$$

Determine (if possible) the zeros of the function \(g\) when the function \(f\) has zeros at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\) $$g(x)=-f(x)$$

Determine (if possible) the zeros of the function \(g\) when the function \(f\) has zeros at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\) $$g(x)=f(-x)$$

The mean salaries \(S\) (in thousands of dollars) of public school classroom teachers in the United States from 2000 through 2011 are shown in the table. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Salary, \(S\) } \\\\\hline 2000 & 42.2 \\\2001 & 43.7 \\\2002 & 43.8 \\\2003 & 45.0 \\\2004 & 45.6 \\\2005 & 45.9 \\\2006 & 48.2 \\\2007 & 49.3 \\\2008 & 51.3 \\\2009 & 52.9 \\\2010 & 54.4 \\\2011 & 54.2 \\\\\hline\end{array}$$ A model that approximates these data is given by $$S=\frac{42.16-0.236 t}{1-0.026 t}, \quad 0 \leq t \leq 11$$ where \(t\) represents the year, with \(t=0\) corresponding to 2000. (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? Explain. (c) Use the model to predict when the salary for classroom teachers will exceed \(\$ 60,000\). (d) Is the model valid for long-term predictions of classroom teacher salaries? Explain.
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