Question 208227: Please help me out with this algebra problem...
I have to graph the function, find the vertex, line of symmetry and max or min. value. I don't really understand it...
f(x)=x^2-8x-1 Thank you for any help!! I really appreciate it!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! equation is x^2 - 8x - 1
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graph the function, find the vertex, line of symmetry and max or min. value.
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this is a quadratic equation in the standard form of ax^2 + bx + c
where a is the coefficient of the x^2 term, b is the coefficient of the x term and c is the constant term
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there are formulas that apply to a quadratic equation.
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the first formula is to find the roots of the equation which is the quadratic formula of:
x = (-b +/- sqrt(b^2 - 4ac))/2a
this formula looks like this:
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the second formula is to find the maximum / minimum point of the equation.
that formula is:
x value of the max/min point = -b/2a
once you find the x value, then you can solve for the y value.
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the axis of symmetry in an open face up quadratic equation or an open face down quadratic equation is a vertical line through the maximum / minimum point.
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to determine whether the quadratic equation will face up or down, take a look at the sign of the x^2 term.
if it is positive, then the quadratic equation opens up.
if it is negative, then the quadratic equation opens down.
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now to your equation.
we'll apply the formulas and see if they work and then we'll graph the equation to visually see what this means.
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your equation is:
y = x^2 - 8x - 1
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a = 1
b = -8
c = -1
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since a quadratic equation produces a parabole, we can call this a parabola.
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since your ax^2 term is positive, this parabola should open up.
this means that the maximum / minimnum point will be a minimum point.
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the equation for the x value of the maximum / minimum point is -b/2a
since a = 1 and b = -8, this equation becomes:
x[min] = - (-8) / 2 = - (-4) = 4
since x = 4, replacing x with 4 in the original equation gets:
y = (4)^2 - 8*4 - 1 = 16 - 32 - 1 = -16 - 1 = -17
your minimun point is (4,-17).
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the axis of symmetry is x = 4 which is a vertical line.
axis of symmetry means that for each value of y you will get 2 values of x that are equidistant from that vertical line.
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let's see how we did.
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graph of y = x^2 - 8x - 1 is shown below:
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you can see that this is an open face up parabola since the 2 tails of the parabola are pointing upward and expanding from each other.
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you can see that the minimun point of the parabola is when x = 4 and y = -17.
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you can see that the axis of symmetry is the vertical line at x = 4.
this takes a little explaining.
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before we do that, we'll take the original equation of y = x^2 - 8x - 1 and solve for x.
the solution is x = 4 +/- sqrt (y + 17)
this formula looks like this:
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when y = 0 this formula produces:
x = 4 +/- 4.123105656 = -.123105656 and + 8.123105656
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you can see on the graph that this is the y axis (when y = 0) and that the graph is showing that these values are correct as far as the naked eye can see.
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any value of y down to just above -17 will produce 2 values for x which are the same distance from x = 4.
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let y = -10
formula becomes:
x = 4 +/- sqrt (-10 + 17) which becomes:
x = 4 +/- sqrt (7) which becmes:
x = 4 +/- 2.645751311
which means that:
x = 1.354248689 or x = 6.645751311
a look at the graph will confirm these values are accurate as far as the naked eye can see.
i put a horizontal line in there at x = -10 and i put 2 vertical lines in there at x = 1.354248689 and x = 6.645751311 to you can see that better.
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hopefully you understand this a little better now.
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a good basic treatment of equation of a parabole can be found at:
http://www.mathwarehouse.com/geometry/parabola/standard-and-vertex-form.php
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there's lots more if you really want to dig in.
just do a search from www.yahoo.com or www.google.com on:
equation of a parabola
axis of symmetry
standard form of equation of parabola
vertex form of equation of parabola
etc.
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