document.write( "Question 821639: I don't know what kind of problem this is...please help!
\n" ); document.write( "Sketch a quadratic function with zeros at -3 and 1. I don't understand the \"zeros at -3 and 1\" part. \n" ); document.write( "
Algebra.Com's Answer #494334 by KMST(5328)\"\" \"About 
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This problem does not require much work, because there are many correct answers.
\n" ); document.write( "You do not need much information or calculations to do that sketch.
\n" ); document.write( "You need information to understand the question.
\n" ); document.write( "Knowing the meaning of some words used by math teachers helps.
\n" ); document.write( "For example a quadratic function is a polynomial function of degree 2, meaning something like
\n" ); document.write( "\"y=3x%5E2%2B5x%2B2\" or \"f%28x%29=x%5E2%2B2x-3\" .
\n" ); document.write( "The graph of a quadratic function is a curve called a parabola, with a more or less pointy \"end\" called the vertex of the parabola.
\n" ); document.write( "It also helps if you know some tricks old mathematicians used to solve similar problems, but you should understand how they figured it out, or at least why those procedures work.
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\n" ); document.write( "One of many functions with zeros at -3 and 1 is
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\n" ); document.write( "It is a quadratic function that
\n" ); document.write( "becomes zero when \"x=1\" because the first factor is zero, and
\n" ); document.write( "becomes zero when \"x=-3\" because the second factor is zero.
\n" ); document.write( "So \"x=1\" and \"x=-3\" are called zeros of the function.
\n" ); document.write( "You could write that function in the factored form,
\n" ); document.write( "\"y=%28x-1%29%28x%2B3%29\"
\n" ); document.write( "or in the standard polynomial form,
\n" ); document.write( "\"y=x%5E2%2B2x-3\"
\n" ); document.write( "or in the vertex form,
\n" ); document.write( "\"y=%28x%2B1%29%5E2-4\" .
\n" ); document.write( "All quadratic functions can be written in the standard form and in the vertex form.
\n" ); document.write( "If the zeros are rational numbers, there is also a nice-looking factored form.
\n" ); document.write( "In that case, it can be proven that the \"x\" at the vertex is the average of the zeros.
\n" ); document.write( "So the \"x\" at the vertex is \"x=%281%2B%28-3%29%29%2F2=%28-2%29%2F2=-1\" ,
\n" ); document.write( "and then \"y\" at the vertex can be found by substituting \"-1\" for \"x\" in one of the forms of the equation for that function.
\n" ); document.write( "
\n" ); document.write( "There are many other quadratic functions with the same zeros,
\n" ); document.write( "because multiplying \"%28x-1%29%28x%2B3%29\" times some non-zero number does not change where the product is zero.
\n" ); document.write( "For example \"y=2%28x-1%29%28x%2B3%29\" is another quadratic function with zeros at -3 and 1. Its graph is \"graph%28200%2C200%2C-5%2C5%2C-8%2C2%2C2x%5E2%2B4x-6%29\"
\n" ); document.write( "So is \"y=-1%28x-1%29%28x%2B3%29=-x%5E2-2x%2B3\" with the graph \"graph%28200%2C200%2C-5%2C5%2C-5%2C5%2C-x%5E2-2x%2B3%29\"
\n" ); document.write( "Multiplying times a factor may change the value of \"y\" at the vertex, but not the zeros or the \"x\" at the vertex.
\n" ); document.write( "So any graph that you sketch looking like that (crossing the x-axis at -3 and 1, and symmetrical to both sides of \"x=-1\" may be a good answer.
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\n" ); document.write( "EXTRA:
\n" ); document.write( "The vertex form is the most useful form of the equation if you want to sketch the graph of a particular function,
\n" ); document.write( "because it tells you the vertex position.
\n" ); document.write( "The minimum value for \"y=%28x%2B1%29%5E2-red%284%29\" is \"y=-red%284%29\" , and happens when \"x=-1\"<-->\"x%2B1=0\"<-->\"%28x%2B1%29%5E2=0\"<-->\"%28x%2B1%29%5E2-4=-4\"
\n" ); document.write( "For all other values of \"x\" , \"%28x%2B1%29%5E2%3E0\" and \"y=%28x%2B1%29%5E2-4%3E-4\" .
\n" ); document.write( "That point, with \"x=-1\" and \"y=-4\" is the vertex of the graph. \n" ); document.write( "
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