document.write( "Question 1208032: How do we find the domain, range and intercepts of any given graph?\r
\n" ); document.write( "\n" ); document.write( "For example, find the domain, range and intercepts of the graph of
\n" ); document.write( "f(x) = x^2 + x - 1? \r
\n" ); document.write( "\n" ); document.write( "NOTE: USE THE GRAPH of the function to explain how this is done VISUALLY. \r
\n" ); document.write( "\n" ); document.write( "NOTE: I don't know how to upload photos in my post. \n" ); document.write( "

Algebra.Com's Answer #846161 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "How do we find the domain, range and intercepts of any given graph?
\n" ); document.write( "For example, find the domain, range and intercepts of the graph of
\n" ); document.write( "f(x) = x^2 + x - 1?
\n" ); document.write( "NOTE: USE THE GRAPH of the function to explain how this is done VISUALLY.
\n" ); document.write( "NOTE: I don't know how to upload photos in my post.
\n" ); document.write( "~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "(1)  The domain of this function is the set of all real numbers.\r\n" );
document.write( "\r\n" );
document.write( "     In other words, the domain is the entire number line, from -infinity to +infinity.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     Thus question (1) is complete.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(2)  This function represents a parabola,\r\n" );
document.write( "\r\n" );
document.write( "     Since the coefficient at x^2 is positive (it is 1 (one)), the parabola has the minimum.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     The range of this function is the set of its values, and it is the set of all real numbers \r\n" );
document.write( "     from the minimum of the parabola to infinity.\r\n" );
document.write( "\r\n" );
document.write( "       \r\n" );
document.write( "     So, to determine the range, we need to determine the minimum.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     The minimum of any quadratic function y = ax^2 + bx + c with positive coefficient \"a\" is at\r\n" );
document.write( "\r\n" );
document.write( "          = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     In our case it is   =  =  = -0.5.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     The minimum is   =  = 0.25 - 0.5 - 1 = -1.25.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     Therefore, the range of this quadratic function is from -1.25 to infinity.\r\n" );
document.write( "\r\n" );
document.write( "     In the interval form, the range is  [-1.25,).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     Thus question (2) is complete.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(3)  To find x-intercepts, you need to solve this equation\r\n" );
document.write( "\r\n" );
document.write( "         x^2 + x - 1 = 0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     Use the quadratic formula\r\n" );
document.write( "\r\n" );
document.write( "          =  = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     The roots are   = =  0.618033989   (approximate),\r\n" );
document.write( "\r\n" );
document.write( "               and   = = -1.618033989   (approximate).\r\n" );
document.write( "\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "At this point, all questions are answered.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To make a plot, go to web-site www.desmos.com/calculator\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Find there free of charge online plotting tool .\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Print there the formula for your function and get the plot instantly.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "////////////////////\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "On finding the maximum/minimum of a quadratic function see the lessons\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the vertex of a parabola\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the vertex of a parabola\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Consider these lessons as your textbook, handbook, tutorials and (free of charge) home teacher, who is always with you to help.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Learn the subject from there once and for all.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );