Final answer:
The quadratic equation f(x) = -2x² + 4x - 7 can be written in vertex form as f(x) = -2(x + 1)² - 5.
Step-by-step explanation:
The quadratic equation f(x) = -2x² + 4x - 7 can be written in vertex form by completing the square. The vertex form of a quadratic equation is given by f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. To convert the equation to vertex form, we need to rewrite it as f(x) = -2(x - h)² + k.
First, let's find the values of h and k. The x-coordinate of the vertex, h, is given by h = -b/2a, where a = -2 and b = 4. So, h = -4/2(-2) = -1.
Next, substitute the value of h into the equation and simplify. f(x) = -2(x - (-1))² + k = -2(x + 1)² + k.
Finally, we need to find the value of k. Substitute any value of x into the equation and solve for f(x). For example, let's substitute x = 0: f(0) = -2(0 + 1)² + k = -2(1) + k = -2 + k.
Since f(0) = -7, we can equate the two expressions: -2 + k = -7. Solving for k, we get k = -5.
Therefore, the quadratic equation f(x) = -2x² + 4x - 7 in vertex form is f(x) = -2(x + 1)² - 5.