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Question 36840: for the funtion f(x)=-x^2+6x-8
(a) Find the line of symmetry
(b) state whether there is a minimum or a maximum and find the value
(c) Find the range
Answer by martinr(6)   (Show Source):
You can put this solution on YOUR website! First, you did not indicate whether you could use a calculator or not, so I will explain it by hand.
First, you know that this graph is a parabola (U-Shaped) due to the fact the the largest exponent it 2. Therefore, there are at most two solutions (points where the graph crosses the x-axis). Also, since the leading coefficient (the number in front of x^2), is negative, you know know that the graph opens down.
Standard form for a quadratic: y = ax^2 + bx + c
Identify a = , b = , and c = . For right now, the c value will not be used in the formulas, but identify it now, because you will use it soon with the quadratic formula.
So a = -1, b = 6 and c = -8.
Find the x-coordinate of the vertex (the turning point of the graph) by using the formula x = -b/2a. This is also the formula for the axis of symmetry (where the graph turns and is a mirror image of itself).
In this case, the x-coordinate of the vertex is x = -6/2(-1) = 3.
To find the y-coordinate, substitute x = 3 into the original equation and find the y-coordinate for the vertex which is -(3)^2 + 6(3) - 8 = 1. The vertes is (3,1).
Now I would choose 2 points greater than the x-coordinate of the vertex and two values less than it as well.
Make a table of values
x y = -x^2 + 6x - 8 y
1 -(1)^2 + 6(1) - 8 = -3
2 -(2)^2 + 6(2) - 8 =0
3 -(3)^2 + 6(3) - 8 =1
4 -(4)^2 + 6(4) - 8 =0
5 -(5)^2 + 6(5) - 8 =-3
Points (1, -3), (2, 0), (3, 1), (4, 0), and (5, -3).
Graph the five points. Remember that this is a U-shaped graph, so there needs to be a curve to it.
The graph is opening down, and the axisof symmetry is a dashed line (vertical) that goes through the vertex.
The maximum value is the vertex (3, 1). It is the highest point that is on the graph.
The range is the y values. Since the graph goes on forever, and it goes negative, therefore the range is highest <= y <= -Infinity. The sign <= means greater than or equal to.
Range: 3<= y <= -Infinity (negative infinity sideways 8).
Hope that this is helpful
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