Given a quadratic function f(x) = 2x^2 - 4x - 3. Answer the question.
a. The vertex (h, k) =?
b. The axis of symmetry?
c. x-intercept(s)?
d. Rewrite f(x) as product of two linear factors with rational constants?
a. To find the vertex (h, k), we can use the formula h = -b/2a for the x-coordinate of the vertex. For our function f(x) = 2x^2 - 4x - 3, we have a = 2 and b = -4. Thus, h = -(-4)/(2*2) = 4/4 = 1. To find the y-coordinate, we can plug h into the function: k = f(1) = 2(1)^2 - 4(1) - 3 = 2 - 4 - 3 = -5. So, the vertex is (1, -5).
b. The axis of symmetry is a vertical line that passes through the vertex. It has the equation x = h, so for our function, the axis of symmetry is x = 1.
c. To find the x-intercepts, we need to find the values of x for which f(x) = 0. So, we need to solve the equation 2x^2 - 4x - 3 = 0. We can use the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a). Plugging in the values for a, b, and c from our function, we get x = (4 ± √((-4)^2 - 4(2)(-3)))/(2(2)) = (4 ± √(16 + 24))/4 = (4 ± √40)/4. So, the x-intercepts are (4 + √40)/4 and (4 - √40)/4.
d. Factoring the given quadratic, we have:
2x^2 - 4x - 3 = 0
It does not factor into rational constants, so we cannot rewrite it as a product of linear factors with rational constants. However, we can write it in the factored form using the x-intercepts we found in part c:
f(x) = 2(x - ((4 + √40)/4))(x - ((4 - √40)/4))