I need a jump start in setting up the equations etc when solving for vertex, line of symmetry and deciding maximum/minimum This is the problem I am starting with
f(x)= -2x^2+2x+8 We are looking for the vertex, line of symmetry and the max/minimum I cannot seem to get this started. Thank you
Rewrite f(x) by completing the square:
f(x) = -2(x^2 - x + 1/4) + 8.5
= -2 (x - 1/2)^2 + 8.5
The vertex is where f(x) has its maximum value. This is where x = 1/2. At that x value, f(x) = 8.5
There is no minimum value.
The function is symmetrical about the x = 1/2 vertical line
ok it says 1-x=y and then a graph its equations
To find the vertex, line of symmetry, and determine whether it is a maximum or minimum, you can follow these steps:
Step 1: Identify the coefficients of the quadratic function.
In your case, the quadratic function is f(x) = -2x^2 + 2x + 8. Here, the coefficient of x^2 is -2, the coefficient of x is 2, and the constant term is 8.
Step 2: Find the x-coordinate of the vertex.
The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a is the coefficient of x^2 and b is the coefficient of x.
In this case, a = -2 and b = 2, so substituting these values into the formula, we get:
x = -2 / (2 * -2) = -2 / -4 = 0.5
Step 3: Find the y-coordinate of the vertex.
To find the y-coordinate of the vertex, substitute the x-coordinate you found in step 2 into the quadratic equation and solve for f(x).
f(0.5) = -2(0.5)^2 + 2(0.5) + 8 = -2(0.25) + 1 + 8 = -0.5 + 1 + 8 = 8.5
So the vertex is (0.5, 8.5).
Step 4: Find the line of symmetry.
The line of symmetry is a vertical line that passes through the vertex. Since the x-coordinate of the vertex is x = 0.5, the equation of the line of symmetry is x = 0.5.
Step 5: Determine whether it is a maximum or minimum.
Since the coefficient of x^2 is -2 (which is negative), the graph of the quadratic function opens downward, meaning it is a maximum. Therefore, the vertex represents the maximum point of the parabola.
In summary, for the quadratic function f(x) = -2x^2 + 2x + 8:
- The vertex is (0.5, 8.5).
- The line of symmetry is x = 0.5.
- The maximum value of f(x) is 8.5.