determine the vortex form and the maximum or minimum value of the function.
f (x) = 2x^2 + 8x +3
To determine the vertex form of the function f(x) = 2x^2 + 8x +3, we need to complete the square.
Step 1: Write the function in the form ax^2 + bx + c
f(x) = 2x^2 + 8x + 3
Step 2: Factor out the coefficient of x^2 from the x^2 and x terms.
f(x) = 2(x^2 + 4x) + 3
Step 3: Complete the square by adding and subtracting (b/2)^2 to the x-term inside the parentheses.
f(x) = 2(x^2 + 4x + (4)^2 - (4)^2) + 3
f(x) = 2[(x + 2)^2 - 4] + 3
f(x) = 2(x + 2)^2 - 8 + 3
f(x) = 2(x + 2)^2 - 5
Now, the vertex form of the function is f(x) = 2(x + 2)^2 - 5.
The vertex of the parabola in the form y = a(x - h)^2 + k is located at the point (h, k).
Therefore, the vertex of the function f(x) = 2x^2 + 8x + 3 is (-2, -5).
To determine whether this vertex represents a maximum or minimum value, we look at the coefficient of x^2. Since the coefficient of x^2 is positive (2), the parabola opens upwards, and the vertex represents the minimum value of the function.
Therefore, the vertex form of the function is f(x) = 2(x + 2)^2 - 5, and the minimum value of the function is -5.