document.write( "Question 1196884: determine the number of intersection points between the two functions below, in terms of the parameter k. Begin with a sketch of the three unique solutions then solve to show how your solutions agrees with your sketch. \r
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Algebra.Com's Answer #829923 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "determine the number of intersection points between the two functions below,
\n" ); document.write( "in terms of the parameter k. Begin with a sketch of the three unique solutions
\n" ); document.write( "then solve to show how your solutions agrees with your sketch.
\n" ); document.write( "y =x (x-8) and y=2x+k
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document.write( "First equation represents a parabola on a coordinate plane, opened up.\r\n" );
document.write( "Second equation represents a straight line with the slope of 2 unit.\r\n" );
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document.write( "The following cases are possible:\r\n" );
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document.write( "    (a) straight line intersects the parabola in two points.\r\n" );
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document.write( "    (b) straight line touches the parabola (having only one common point with the parabola).\r\n" );
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document.write( "    (c) straight line is located below the parabola and does not have intersection points with the parabola.\r\n" );
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document.write( "To analyze these cases, we consider the system of equations\r\n" );
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document.write( "    y = x*(x-8)    (1)\r\n" );
document.write( "    y = 2x + k     (2)\r\n" );
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document.write( "From equations (1) and (2), we have this equation\r\n" );
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document.write( "    x*(x-8) = 2x +k.\r\n" );
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document.write( "Simplify it\r\n" );
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document.write( "    x^2 - 8x = 2x + k,\r\n" );
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document.write( "    x^2 - 10x - k = 0.\r\n" );
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document.write( "The last equation is a standard form quadratic equation.\r\n" );
document.write( "All the cases are controlled by the discriminant of this quadratic equation.\r\n" );
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document.write( "Referring to the general form quadratic equation, the discriminant is\r\n" );
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document.write( "    d = b^2 - 4ac.\r\n" );
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document.write( "In your problem,\r\n" );
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document.write( "    d = (-10)^2 - 4*1*(-k) = 100 + 4k.\r\n" );
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document.write( "Case (a) (two solutions and two intersection points) may happen if and only if the discriminant is positive\r\n" );
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document.write( "    100+4k > 0,  which implies  k > -100/4 = -25.\r\n" );
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document.write( "Case (b) (one solution, touching) may happen if and only if the discriminant is zero\r\n" );
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document.write( "    100+4k = 0,  which implies  k = -100/4 = -25.\r\n" );
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document.write( "Case (c) (no solutions, no intersection points) may happen if and only if the discriminant is negative\r\n" );
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document.write( "    100+4k < 0,  which implies  k < -100/4 = -25.\r\n" );
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document.write( "Thus, you have this ANSWER\r\n" );
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document.write( "    - there are two intersection point if and only if k > -25;\r\n" );
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document.write( "    - there is only one common point (touching) if and only if k = -25;\r\n" );
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document.write( "    - there are no intersection points if and only if k < -25.\r\n" );
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document.write( "To get plots, go to web-site www.desmos.com/calculator\r\n" );
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document.write( "and use free of charge plotting tool there.\r\n" );
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document.write( "Print your given equations there, taking appropriate numerical value of k (k= -30;  k= -25;  k= -20)\r\n" );
document.write( "and obtain the required plots momentarily.\r\n" );
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