Find the line of symmetry, the maximum or minimum value of the quadratic function, and graph the function f(x)=-2x^2+2x+9
Y = -2X^2 + 2X +9.
Vertex form: a( X - h)^2 + k.
h = Xv = -b/2a = -2/-4 = 1/2
k = Yv = -2(1/2)^2 + 2(1/2) + 9 = 19/2
V(h,k) = (1/2,19/2). a is negative; the
parabola opens downward. Therefore, the
vertex is a max. The line of symmetry is = h = 1/2.
To find the line of symmetry, we can use the formula x = -b / 2a, where the quadratic equation is in the form of f(x) = ax^2 + bx + c.
In this case, the given quadratic equation is f(x) = -2x^2 + 2x + 9. Comparing this equation with the general form, we can see that a = -2, b = 2, and c = 9.
Using the formula for the line of symmetry, x = -b / 2a, we can substitute the values to find the line of symmetry:
x = -(2) / 2(-2)
x = -2 / -4
x = 0.5
Therefore, the line of symmetry for the given quadratic function is x = 0.5.
Next, to find the maximum or minimum value of the quadratic function, we can substitute the x-coordinate of the line of symmetry back into the original equation.
Substituting x = 0.5 into f(x) = -2x^2 + 2x + 9:
f(0.5) = -2(0.5)^2 + 2(0.5) + 9
f(0.5) = -2(0.25) + 1 + 9
f(0.5) = -0.5 + 1 + 9
f(0.5) = -0.5 + 10
f(0.5) = 9.5
So, the maximum or minimum value of the quadratic function f(x) = -2x^2 + 2x + 9 is 9.5.
To graph the quadratic function, we can plot some points and sketch the curve.
By choosing a few x-values, we can calculate the corresponding y-values. Let's choose the values -2, -1, 0, 1, and 2.
x = -2: f(-2) = -2(-2)^2 + 2(-2) + 9 = -8 - 4 + 9 = -3
x = -1: f(-1) = -2(-1)^2 + 2(-1) + 9 = -2 + 2 + 9 = 9
x = 0: f(0) = -2(0)^2 + 2(0) + 9 = 9
x = 1: f(1) = -2(1)^2 + 2(1) + 9 = -2 + 2 + 9 = 9
x = 2: f(2) = -2(2)^2 + 2(2) + 9 = -8 + 4 + 9 = 5
Plotting these points (-2, -3), (-1, 9), (0, 9), (1, 9), and (2, 5), we can then sketch the graph of the function f(x) = -2x^2 + 2x + 9.
To find the line of symmetry and the maximum or minimum value of a quadratic function in the form f(x) = ax^2 + bx + c, we can use the formula x = -b / (2a) for the line of symmetry, and substitute the value of x into the equation to find the corresponding y-value, which will give us the maximum or minimum.
In the given quadratic function f(x) = -2x^2 + 2x + 9, we can identify that a = -2, b = 2, and c = 9.
1. Line of Symmetry:
Using the formula x = -b / (2a), we substitute the values to find the line of symmetry:
x = -(2) / (2*(-2)) = -2 / (-4) = 1/2
So, the line of symmetry is x = 1/2.
2. Maximum or Minimum Value:
To find the maximum or minimum value, we substitute the line of symmetry x = 1/2 into the equation:
f(1/2) = -2(1/2)^2 + 2(1/2) + 9
Simplifying,
f(1/2) = -2(1/4) + 1 + 9
f(1/2) = -1/2 + 10/2
f(1/2) = 9.5
Therefore, the quadratic function f(x) = -2x^2 + 2x + 9 has a maximum value of 9.5.
3. Graphing the Function:
To graph the quadratic function, we can either use a graphing calculator or manually plot points. Here is a table of some x and y values:
x | y
-2 | 9
-1 | 7
0 | 9
1 | 9
2 | 7
Now we can plot these points on the coordinate plane and join them to form the graph of the quadratic function f(x) = -2x^2 + 2x + 9. The line of symmetry will pass through the vertex, which in this case is at x = 1/2 and y = 9.5. The graph will be a downward-opening parabola symmetric to the line of symmetry x = 1/2.