91Ó°ÊÓ

The graph of \(f(x)=|x|\) is transformed to the graph of \(g(x)=f(x-9)+5\). a) Determine the equation of the function \(g(x)\) b) Compare the graph of \(g(x)\) to the graph of the base function \(f(x)\) c) Determine three points on the graph of \(f(x) .\) Write the coordinates of the image points if you perform the horizontal translation first and then the vertical translation. d) Using the same original points from part c), write the coordinates of the image points if you perform the vertical translation first and then the horizontal translation. e) What do you notice about the coordinates of the image points from parts \(c\) ) and \(d\) )? Is the order of the translations important?

An object falling in a vacuum is affected only by the gravitational force. An equation that can model a free-falling object on Earth is \(d=-4.9 t^{2},\) where \(d\) is the distance travelled, in metres, and \(t\) is the time, in seconds. An object free falling on the moon can be modelled by the equation \(d=-1.6 t^{2}\) a) Sketch the graph of each function. b) Compare each function equation to the base function \(d=t^{2}\)

For each of the following functions,determine the equation of the inverse graph \(f(x)\) and the inverse of \(f(x)\) restrict the domain of \(f(x)\) so that the inverse of \(f(x)\) is a function with the domain of \(f(x)\) restricted, sketch the graphs of \(f(x)\) and \(f^{-1}(x)\). a) \(f(x)=x^{2}+3\) b) \(f(x)=\frac{1}{2} x^{2}\) c) \(f(x)=-2 x^{2}\) d) \(f(x)=(x+1)^{2}\) e) \(f(x)=-(x-3)^{2}\) f) \(f(x)=(x-1)^{2}-2\)

The speed of a vehicle the moment the brakes are applied can be determined by its skid marks. The length, \(D\), in feet, of the skid mark is related to the speed, \(S\) in miles per hour, of the vehicle before braking by the function \(D=\frac{1}{30 f n} S^{2},\) where \(f\) is the drag factor of the road surface and \(n\) is the braking efficiency as a decimal. Suppose the braking efficiency is \(100 \%\) or 1 a) Sketch the graph of the length of the skid mark as a function of speed for a drag factor of \(1,\) or \(D=\frac{1}{30} S^{2}\) b) The drag factor for asphalt is \(0.9,\) for gravel is \(0.8,\) for snow is \(0.55,\) and for ice is 0.25. Compare the graphs of the functions for these drag factors to the graph in part a).

Gil is asked to translate the graph of \(y=|x|\) according to the equation \(y=|2 x-6|+2\) He decides to do the horizontal translation of 3 units to the right first, then the stretch about the \(y\) -axis by a factor of \(\frac{1}{2},\) and lastly the translation of 2 units up. This gives him Graph 1. To check his work, he decides to apply the horizontal stretch about the \(y\) -axis by a factor of \(\frac{1}{2}\) first, and then the horizontal translation of 6 units to the right and the vertical translation of 2 units up. This results in Graph 2. a) Explain why the two graphs are in different locations. b) How could Gil have rewritten the equation so that the order in which he did the transformations for Graph 2 resulted in the same position as Graph \(1 ?\)

For each function, state two ways to restrict the domain so that the inverse is a function a) \(f(x)=x^{2}+4\) b) \(f(x)=2-x^{2}\) b) \(f(x)=2-x^{2}\) c) \(f(x)=(x-3)^{2}\) d)\(f(x)=(x+2)^{2}-4\)

Two parabolic arches are being built. The first arch can be modelled by the function \(y=-x^{2}+9,\) with a range of \(0 \leq y \leq 9\) The second arch must span twice the distance and be translated 6 units to the left and 3 units down. a) Sketch the graph of both arches. b) Determine the equation of the second arch.

Consider the function \(f(x)=(x+4)(x-3)\) Without graphing, determine the zeros of the function after each transformation. a) \(y=4 f(x)\) b) \(y=f(-x)\) c) \(y=f\left(\frac{1}{2} x\right)\) d) \(y=f(2 x)\)

If the \(x\) -intercept of the graph of \(y=f(x)\) is located at \((a, 0)\) and the \(y\) -intercept is located at \((0, b),\) determine the \(x\) -intercept and \(y\) -intercept after the following transformations of the graph of \(y=f(x)\). a) \(y=-f(-x)\) b) \(y=2 f\left(\frac{1}{2} x\right)\) c) \(y+3=f(x-4)\) d) \(y+3=\frac{1}{2} f\left(\frac{1}{4}(x-4)\right)\)

Given the function \(f(x)=4 x-2\) determine each of the following. a) \(f^{-1}(4)\) b) \(f^{-1}(-2)\) c) \(f^{-1}(8)\) d) \(f^{-1}(0)\)