document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165384>Question 1165384</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The graph of f(x) is shown below. https://latex.artofproblemsolving.com/d/b/a/dba24dce7c2734af95b18f61b73b2a32f213cd79.png<BR>\n" );
document.write( "For each point (a,b) on the graph of y = f(x), the point ( 3a - 1, b/2) is plotted to form the graph of another function y = g(x). For example, (0,2) lies on the graph of y = f(x), so (3 * 0 - 1, 2/2) = (-1,1) lies on the graph of y = g(x).\r<BR>\n" );
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document.write( "(a) Plot the graph of y = g(x). Include the diagram in your solution.\r<BR>\n" );
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document.write( "(b) Express g(x) in terms of f(x).\r<BR>\n" );
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document.write( "(c) Describe the transformations that you would apply to the graph of y = f(x) to obtain the graph of y = g(x). For example, one transformation might be to stretch the graph horizontally by a factor of 5. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853098 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=853098\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem involves understanding function transformations based on how the coordinates of points change.<BR>\n" );
document.write( "Since the graph of $f(x)$ is not provided, I will define a simple, representative graph with clear points to illustrate the transformations.<BR>\n" );
document.write( "## Example Graph: $y = f(x)$<BR>\n" );
document.write( "Let's assume the graph of $f(x)$ has the following four distinct points, which define its shape :<BR>\n" );
document.write( "* $P_1 = (-4, 4)$<BR>\n" );
document.write( "* $P_2 = (0, -2)$<BR>\n" );
document.write( "* $P_3 = (4, 2)$<BR>\n" );
document.write( "---<BR>\n" );
document.write( "## (a) Plot the graph of $y = g(x)$<BR>\n" );
document.write( "The transformation rule is: Every point $(a, b)$ on the graph of $y = f(x)$ becomes the point $(5a, b - 3/2)$ on the graph of $y = g(x)$.<BR>\n" );
document.write( "Applying the transformation $(x, y) \to \left(5x, y - \frac{3}{2}\right)$ to our example points:<BR>\n" );
document.write( "* $P_1(-4, 4) \to P'_1 \left(5(-4), 4 - \frac{3}{2}\right) = P'_1 (-20, 2.5)$<BR>\n" );
document.write( "* $P_2(0, -2) \to P'_2 \left(5(0), -2 - \frac{3}{2}\right) = P'_2 (0, -3.5)$<BR>\n" );
document.write( "* $P_3(4, 2) \to P'_3 \left(5(4), 2 - \frac{3}{2}\right) = P'_3 (20, 0.5)$<BR>\n" );
document.write( "The graph of $y = g(x)$ connects the points $(-20, 2.5)$, $(0, -3.5)$, and $(20, 0.5)$.\r<BR>\n" );
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document.write( "---<BR>\n" );
document.write( "## (b) Express $g(x)$ in terms of $f(x)$<BR>\n" );
document.write( "To find the function $y = g(x)$, we reverse the transformations applied to the coordinates.<BR>\n" );
document.write( "Let $(X, Y)$ be a point on the graph of $g(x)$, and let $(x, y)$ be the corresponding point on the graph of $f(x)$.<BR>\n" );
document.write( "The transformation rule is:<BR>\n" );
document.write( "1. $X = 5x$<BR>\n" );
document.write( "2. $Y = y - \frac{3}{2}$<BR>\n" );
document.write( "We need to solve for $x$ and $y$ in terms of $X$ and $Y$:<BR>\n" );
document.write( "1. $x = \frac{1}{5} X$<BR>\n" );
document.write( "2. $y = Y + \frac{3}{2}$<BR>\n" );
document.write( "Since $(x, y)$ is on the graph of $f(x)$, we have $y = f(x)$. Substitute the expressions for $x$ and $y$:<BR>\n" );
document.write( "$$Y + \frac{3}{2} = f\left(\frac{1}{5} X\right)$$<BR>\n" );
document.write( "Now, solve for $Y$:<BR>\n" );
document.write( "$$Y = f\left(\frac{1}{5} X\right) - \frac{3}{2}$$<BR>\n" );
document.write( "Replacing $Y$ with $g(X)$ and using $x$ as the standard variable:<BR>\n" );
document.write( "$$\mathbf{g(x) = f\left(\frac{1}{5} x\right) - \frac{3}{2}}$$<BR>\n" );
document.write( "---<BR>\n" );
document.write( "## (c) Describe the transformations<BR>\n" );
document.write( "The expression $g(x) = f\left(\frac{1}{5} x\right) - \frac{3}{2}$ reveals two transformations applied to $f(x)$:<BR>\n" );
document.write( "1. **Horizontal Transformation (due to $\frac{1}{5}x$ inside $f$):**<BR>\n" );
document.write( "The $x$-coordinates are multiplied by 5, which corresponds to a **horizontal stretch by a factor of 5**.<BR>\n" );
document.write( "$$\left(x \to \frac{1}{5}x\right) \implies \text{Horizontal Stretch by } 5$$<BR>\n" );
document.write( "2. **Vertical Transformation (due to $-\frac{3}{2}$ outside $f$):**<BR>\n" );
document.write( "The constant $\frac{3}{2}$ is subtracted from the $y$-value, which corresponds to a **vertical shift downward by $\frac{3}{2}$ units (or $1.5$ units)**.<BR>\n" );
document.write( "$$\left(y \to y - \frac{3}{2}\right) \implies \text{Vertical Shift Down by } \frac{3}{2}$$  </SPAN>\n" );
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