document.write( "Question 753632: Without graphing, find the vertex and the maximum and minimum value of f(x)
\n" ); document.write( "f(x)= -4/7(x-4)^2+8 \n" ); document.write( "

Algebra.Com's Answer #458588 by KMST(5328) $\"\"$ $\"About$

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$\"f%28x%29=+%28-4%2F7%29%28x-4%29%5E2%2B8\"$ is a quadratic function, so its graph is a parabola.
\n" ); document.write( " $\"f%28x%29=+%28-4%2F7%29%28x-4%29%5E2%2B8\"$ is the equation of that parabola in vertex form.
\n" ); document.write( "The vertex form is the form that makes it easiest to find the vertex.
\n" ); document.write( "When $\"x=4\"$ $\"x-4=0\"$ and $\"f%28x%29=+%28-4%2F7%29%28x-4%29%5E2%2B8=%28-4%2F7%290%5E2%2B8=8\"$
\n" ); document.write( "For all other values of $\"x\"$ ,
\n" ); document.write( " $\"%28x-4%29%5E2%3E0\"$ , $\"%28-4%2F7%29%28x-4%29%5E2%3C0\"$ , and $\"f%28x%29=+%28-4%2F7%29%28x-4%29%5E2%2B8%3C=8\"$
\n" ); document.write( "No matter what value $\"x\"$ takes, $\"f%28x%29%3C=8\"$
\n" ); document.write( "and $\"f%28x%29=8\"$ only when $\"x=4\"$ .
\n" ); document.write( "The function has a maximum at $\"x=4\"$ ,
\n" ); document.write( "and the value of that maximum is $\"f%284%29=8\"$ .
\n" ); document.write( "That corresponds to the vertex of the parabola, the point (4,8).
\n" ); document.write( "The function does not have a minimum; it can take any negative value you can think of.
\n" ); document.write( "The graph looks like this:
\n" ); document.write( " $\"graph%28300%2C300%2C-6%2C14%2C-10%2C10%2C%28-4%2F7%29%28x-4%29%5E2%2B8%29\"$
\n" ); document.write( "Parabolas can look like this $\"graph%28100%2C100%2C1%2C7%2C3%2C9%2C%28-4%2F7%29%28x-4%29%5E2%2B8%29\"$ or like this $\"graph%28100%2C100%2C1%2C7%2C-9%2C-3%2C4%2F7%2A%28x-4%29%5E2-8%29\"$
\n" ); document.write( "They can have a maximum or a minimum, but not both, and whichever they have happens at the vertex. \n" ); document.write( "

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