document.write( "Question 1208030: For each point (a,b) that is on the graph of y = f(x), the point (2a +1, 3b) is plotted, forming the graph of another function y = g(x). As an example, the point (0,2) lies on the graph of y = f(x), so the point (2 • 0 + 1, 3 • 2) = (1, 6) lies on the graph of y = g(x).\r
\n" ); document.write( "\n" ); document.write( "(a) Plot the graph of y = g(x). Include the diagram as part of your solution.\r
\n" ); document.write( "\n" ); document.write( "(b) Express g(x) in terms of f(x).\r
\n" ); document.write( "\n" ); document.write( "(c) Describe the transformations that can be applied to the graph of y = f(x) to obtain the graph of y = g(x). For example, one transformation could be to stretch the graph vertically by a factor of 4.\r
\n" ); document.write( "\n" ); document.write( "The graph: https://artofproblemsolving.com/texer/gbtdafpj \n" ); document.write( "

Algebra.Com's Answer #846157 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "For each point (a,b) that is on the graph of y = f(x), the point (2a +1, 3b) is plotted,
\n" ); document.write( "forming the graph of another function y = g(x).
\n" ); document.write( "As an example, the point (0,2) lies on the graph of y = f(x), so the point
\n" ); document.write( "(2 • 0 + 1, 3 • 2) = (1, 6) lies on the graph of y = g(x).
\n" ); document.write( "(a) Plot the graph of y = g(x). Include the diagram as part of your solution.
\n" ); document.write( "(b) Express g(x) in terms of f(x).
\n" ); document.write( "(c) Describe the transformations that can be applied to the graph of y = f(x) to obtain the graph of y = g(x).
\n" ); document.write( "For example, one transformation could be to stretch the graph vertically by a factor of 4.
\n" ); document.write( "The graph: https://artofproblemsolving.com/texer/gbtdafpj
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " I will solve (b) and (c). You make the rest.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "(b)  As the problem describes, we start from the point (a,b) = (a,f(a)).\r\n" );
document.write( "     \r\n" );
document.write( "     We transform it into the point (2a+1,g) = (2a+1,3f(a)).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     So,  g(2a+1) = 3*f(a).    (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     Now we want to call 2a+1 as new variable x;  so we write\r\n" );
document.write( "\r\n" );
document.write( "           x = 2a + 1,  which means that  a = .\r\n" );
document.write( "\r\n" );
document.write( "  \r\n" );
document.write( "     Then the equality (1) takes the form\r\n" );
document.write( "\r\n" );
document.write( "            g(x) = .    (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     This formula (2) defines function g(x) via function f(x).\r\n" );
document.write( "\r\n" );
document.write( "      So, formula (2) is the answer to question (b).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "      At this point, part (b) is complete.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(c)  Based on formula (2) from part (b), the transformations that can be applied to the graph of y = f(x) \r\n" );
document.write( "     to obtain the graph of y = g(x)  are as follows:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "         - shift the plot of f(x) 1 unit right;\r\n" );
document.write( "\r\n" );
document.write( "         - then stretch x-axis with coefficient 2;\r\n" );
document.write( "\r\n" );
document.write( "         - then stretch y-axis with coefficient 3.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Parts (b) and (c) are completed.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );