find the vertex line of symmetry the minimum naximum value of the quadratic function and graph the function f(x)=-2x^2+2x+2.
A parabola has either a minimum or a maximum, but not both.
The axis of symmetry, which contains the vertex, is the line
x = -b/2a = -2/(-4) = 1/2
You should now be able to get the max/min and thus the vertex
Express the function in canonical form by completing squares:
if f(x)=a(x-h)²+k,
then
a>0 => the parabola is concave up, hence a minimum exists
a<0 => concave down, hence a maximum.
The location of maximum/minimum is given by the point (h,k).
The line of symmetry is x=h.
To proceed with completing the squares, extract and factor out the coefficient of x²:
f(x)=-2(x²-x) + 2
=-2[(x-1/2)²-(1/2)²]+2
=-2[(x-1/2)²]+5/2
So the curve is concave down, h=1/2, k=5/2 and (h,k)=(1/2,5/2) is a maximum.
The line of symmetry is x=1/2.
How do I find the Y coordinate and the x coordinates
is the vertix 1/2, 5/2?
By completing squares, we get the two parameters h and k which represent the x- and y-coordinates of the vertex.
In this case, they are (1/2,5/2), as you can see from:
f(x) = -2[(x-1/2)²]+5/2
To find the vertex, line of symmetry, minimum/maximum value of a quadratic function, and graph the function f(x) = -2x^2 + 2x + 2, follow these steps:
Step 1: Find the Vertex
The vertex of a quadratic function is given by the formula: x = -b / (2a), where a, b, and c are coefficients of the function in the form: ax^2 + bx + c.
In our case, a = -2 and b = 2. So, substituting these values into the formula, we get:
x = -2 / (2*(-2))
x = -2 / (-4)
x = 0.5
To find the corresponding y-coordinate, substitute the x-value back into the function:
f(0.5) = -2(0.5)^2 + 2(0.5) + 2
f(0.5) = -2(0.25) + 1 + 2
f(0.5) = -0.5 + 3
f(0.5) = 2.5
Therefore, the vertex is (0.5, 2.5).
Step 2: Line of Symmetry
The line of symmetry of a quadratic function is the vertical line that passes through its vertex. In this case, the line of symmetry is x = 0.5.
Step 3: Minimum/Maximum Value
For the quadratic function, the vertex represents either the minimum or maximum value, depending on whether the coefficient of x^2 (a) is positive or negative.
In our case, a = -2 (which is negative), so the parabola opens downwards, and the vertex represents the maximum value.
Therefore, the maximum value of the function is 2.5.
Step 4: Graphing
To graph the quadratic function, plot the vertex (0.5, 2.5) on the coordinate plane. Since the parabola opens downwards, the graph will be a downward-facing "U" shape.
You can also find additional points on the graph by plugging in other x-values into the equation and evaluating the corresponding y-values. For example, if we choose x = -1, we can find the y-value:
f(-1) = -2(-1)^2 + 2(-1) + 2
f(-1) = -2(1) - 2 + 2
f(-1) = -2 - 2 + 2
f(-1) = -2
So, another point on the graph is (-1, -2).
You can go ahead and plot more points by following the same process. Once you have a few points, connect them smoothly to create the graph of the quadratic function f(x) = -2x^2 + 2x + 2. The graph should be a downward-facing parabola passing through the vertex at (0.5, 2.5).
I hope this helps you understand how to find the vertex, line of symmetry, minimum/maximum value, and graph a quadratic function!