Question 166503: PLEASE HELP!!!
Graph f(x)=-x^2+4x-3, labeling the y intercept, vertex, and axis of symmetry.
Also
graph f(x)=(-x)^4+4x^2-2x by making a table of values. Then estimate the x-coordinates at which the relative maxima and relative minima occur.
I must show my work. This is a worksheet.
Thanks
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! first part:
Graph f(x)=-x^2+4x-3, labeling the y intercept, vertex, and axis of symmetry.
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equation is f(x) = -x^2 + 4x -3
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since there is a minus sign in front of the x^2, this equation will point up and open down.
this means that the vertex will be a maximum point.
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since this equation is in the form of:
a*x^2 + b*x + c,
the formula for the maximum / minimum point of the graph is:
x = -b/2a
this point will be a maximum point.
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b = 4
a = -1
-b / 2a = (-4)/(-2) = 2
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the x value of the maximum point of the graph is:
x = 2
the y value of the maximum point of the graph is found by substituting x = 2 in the equation.
the equation is:
f(x) = -x^2 + 4x -3
substituting 2 for x makes the equation become:
f(2) = -(2)^2 + 4(2) - 3
this becomes:
- 4 + 8 - 3 = 1
the y value of the maximum point is 1.
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the coordinates of the maximum point are:
x = 2
y = 1
coordinates (x,y) are: (2,1)
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the vertex is the maximum point of this graph.
the x value of the vertex is the line of symmetry.
line of symmetry = x = 2
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the y intercept of this graph is found by making x = 0
f(0) = -(0)^2 + 4*0 - 3
f(0) = -3
y intercept of this graph is -3.
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the graph looks like this:
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as you can see, the y intercept is -3 (value of y when x = 0).
the vertex is the maximum point of the graph = (2,1)
the axis of symmetry = x = 2.
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second part:
graph f(x)=(-x)^4+4x^2-2x by making a table of values. Then estimate the x-coordinates at which the relative maxima and relative minima occur.
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this graph looks like it will open upwards and point downward since all values of f(x) look like they will be positive.
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to prove this: plot of 7 points should be more than sufficient.
the points to be plotted are
(x,y)
(-3,123)
(-2,36)
(-1,7)
(0,0)
(1,3)
(2,28)
(3,111)
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the x coordinates of the relative minima are 0.
there are no maxima.
graph will open upwards and continue to infinity.
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graph of this equation looks like:
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