91Ó°ÊÓ

Let \(f(x)=x^{3}\). a. Write the equation for the new function \(g(x)\) that results from each of the following transformations of \(f(x)\). Explain in words the effect of the transformations. i. \(f(-x)\) iii. \(f(x+2)\) ii. \(-2 f(x)-1\) iv. \(-f(-x)\) b. Sketch by hand the graph of \(f(x)\) and each function in part (a).

Identify the stretch/compression factor and the vertex for each of the following. a. \(y_{1}=0.3(x-1)^{2}+8\) c. \(y_{3}=0.01(x+20)^{2}\) b. \(y_{2}=30 x^{2}-11\) d. \(y_{4}=-6 x^{2}+12 x\)

Find the coordinates of the vertex for each quadratic function listed. Then specify whether each vertex is a maximum or minimum. a. \(y=4 x^{2}\) c. \(P(n)=\left(\frac{1}{12}\right) n^{2}\) b. \(f(x)=-8 x^{2}\) d. \(Q(t)=-\left(\frac{1}{24}\right) t^{2}\)

Write each function in factored form, if possible, using integer coefficients. a. \(f(x)=x^{2}+2 x-15\) d. \(k(p)=p^{2}+5 p+7\) b. \(g(x)=x^{2}-6 x+9\) e. \(l(s)=5 s^{2}-37 s-24\) c. \(h(a)=a^{2}+6 a-16\) f. \(m(t)=5 t^{2}+t+1\)

For each of the following quadratic functions, find the vertex \((h, k)\) and determine if it represents the maximum or minimum of the function. a. \(f(x)=-2(x-3)^{2}+5\) c. \(f(x)=-5(x+4)^{2}-7\) b. \(f(x)=1.6(x+1)^{2}+8\) d. \(f(x)=8(x-2)^{2}-6\)

(Graphing program optional.) Create a quadratic function in the vertex form \(y=a(x-h)^{2}+k,\) given the specified values for \(a\) and the vertex \((h, k) .\) Then rewrite the function in the standard form \(y=a x^{2}+b x+c .\) If available, use technology to check that the graphs of the two forms are the same. a. \(a=1,(h, k)=(2,-4)\) c. \(a=-2,(h, k)=(-3,1)\) b. \(a=-1,(h, k)=(4,3)\) d. \(a=\frac{1}{2},(h, k)=(-4,6)\)

For each of the following functions, evaluate \(f(2)\) and \(f(-2)\). a. \(f(x)=x^{2}-5 x-2\) b. \(f(x)=3 x^{2}-x\) c. \(f(x)=-x^{2}+4 x-2\)

The wind chill temperature is the apparent temperature caused by the extra cooling from the wind. A rule of thumb for estimating the wind chill temperature for an actual temperature \(t\) that is above \(0^{\circ}\) Fahrenheit is \(W(t)=t-1.5 S_{0}\), where \(S_{0}\) is any given wind speed in miles per hour. a. If the wind speed is 25 mph and the actual temperature is \(10^{\circ} \mathrm{F}\), what is the wind chill temperature? We know how to convert Celsius to Fahrenheit; that is, we can write \(t=F(x),\) where \(F(x)=32+\frac{9}{5} x,\) with \(x\) the number of degrees Celsius and \(F(x)\) the equivalent in degrees Fahrenheit. b. Construct a function that will give the wind chill temperature as a function of degrees Celsius. c. If the wind speed is \(40 \mathrm{mph}\) and the actual temperature is \(-10^{\circ} \mathrm{C},\) what is the wind chill temperature?

For the following quadratic functions in vertex form, \(f(x)=a(x-h)^{2}+k,\) determine the values for \(a, h,\) and \(k\) Then compare each to \(f(x)=x^{2},\) and identify which constants represent a stretch/compression factor, or a shift in a particular direction. a. \(p(x)=5(x-4)^{2}-2\) b. \(g(x)=\frac{1}{3}(x+5)^{2}+4\) c. \(h(x)=-0.25\left(x-\frac{1}{2}\right)^{2}+6\) d. \(k(x)=-3(x+4)^{2}-3\)

If we know the radius and depth of a parabolic reflector, we also know where the focus is. a. Find a generic formula for the focal length \(f\) of a parabolic reflector expressed in terms of its radius \(R\) and depth \(D\). The focal length \(\left|\frac{1}{4 a}\right|\) is the distance between the vertex and the focal point. Assume \(a>0\). b. Under what conditions does \(f=D ?\)