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Chapter 2: Problem 22

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$f(x)=x^{2}-12 x+44$$

Short Answer

Expert verified
The quadratic function in standard form is \(f(x) = (x-6)^2 + 8\). The vertex is (6, 8), and the axis of symmetry is \(x=6\). The function has no x-intercepts.

Step by step solution

01

Rewrite in Standard Form

The given quadratic function is \(f(x) = x^2 - 12x + 44\). This can be written in the standard form by completing the square. Rearrange the equation such that the square of a binomial is formed: \(f(x) = (x^2 - 12x + 36) + 44 - 36 -> f(x) = (x-6)^2 + 8\). So the equation in standard form is \(f(x) = (x-6)^2 + 8\).
02

Identify the Vertex and Axis of Symmetry

The vertex of the function can be identified directly from standard form. In this case, \(h=6\) and \(k=8\), so the vertex is (6, 8). The axis of symmetry is \(x=h\), which is \(x=6\).
03

Identify the x-Intercept(s)

The x-intercepts can be found by setting \(f(x) = 0\) and solving for \(x\). In this case, we have \(0 = (x-6)^2 + 8\). Solving this, we subtract 8 from both sides, then take the square root. We obtain \(x = 6 + sqrt(-8)\) and \(x = 6 - sqrt(-8)\). Since square roots of negative numbers are complex and x-intercepts must be real numbers, this quadratic function has no x-intercepts.

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Most popular questions from this chapter

The numbers \(N\) (in millions) of students enrolled in schools in the United States from 2000 through 2009 are shown in the table. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Number, \(N\) } \\\\\hline 2000 & 72.2 \\\2001 & 73.1 \\\2002 & 74.0 \\\2003 & 74.9 \\\2004 & 75.5 \\\2005 & 75.8 \\\2006 & 75.2 \\\2007 & 76.0 \\\2008 & 76.3 \\\2009 & 77.3 \\\\\hline\end{array}$$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=0\) corresponding to 2000. (b) Use the regression feature of the graphing utility to find a quartic model for the data. (A quartic model has the form \(a t^{4}+b t^{3}+c t^{2}+d t+e,\) where \(a, b\) \(c, d, \text { and } e \text { are constant and } t \text { is variable. })\) (c) Graph the model and the scatter plot in the same viewing window. How well does the model fit the data? (d) According to the model, when did the number of students enrolled in schools exceed 74 million? (e) Is the model valid for long-term predictions of student enrollment? Explain.

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$h(x)=2 x^{4}-3 x+2$$

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=2 x^{4}-8 x+3\) (a) Upper: \(x=3\) (b) Lower: \(x=-4\)

Use the given zero to find all the zeros of the function. Function \(g(x)=x^{3}-7 x^{2}-x+87\) Zero \(5+2 i\)

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$x^{2}+b x-4=0$$
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