Chapter 4: Problem 53
53\. Simplify: \(\frac{5 x^{4}(2 x+7)^{4}-8 x^{5}(2 x+7)^{3}}{(2 x+7)^{8}}\)
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$$\text { If } g(x)=\frac{2}{3} x-8, \text { find } g\left(\frac{3}{2} x+12\right)$$
Use the fact that a quadratic function of the form \(f(x)=a x^{2}+b x+c\) with \(b^{2}-4 a c>0\) may also be written in the form \(f(x)=a\left(x-r_{1}\right)\left(x-r_{2}\right),\) where \(r_{1}\) and \(r_{2}\) are the \(x\) -intercepts of the graph of the quadratic function. (a) Find quadratic functions whose \(x\) -intercepts are -5 and 3 with \(a=1 ; a=2 ; a=-2 ; a=5\) (b) How does the value of \(a\) affect the intercepts? (c) How does the value of \(a\) affect the axis of symmetry? (d) How does the value of \(a\) affect the vertex? (e) Compare the \(x\) -coordinate of the vertex with the midpoint of the \(x\) -intercepts. What might you conclude?
Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. $$ \text { Graph } g(x)=\left\\{\begin{array}{cl} x^{2} & \text { if } x \leq 0 \\ \sqrt{x}+1 & \text { if } x>0 \end{array}\right. $$
(a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the \(x\) -intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where \(f(x)>0\) and where \(f(x)<0\) \(f(x)=x^{2}-2 x-3\)
Maximizing Revenue Suppose that the manufacturer of a gas clothes dryer has found that when the unit price is \(p\) dollars, the revenue \(R\) (in dollars) is $$ R(p)=-4 p^{2}+4000 p $$ What unit price \(p\) maximizes revenue? What is the maximum revenue?
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