f(x)=x^2-2x

Let's analyze the function \( f(x) = x^2 - 2x \). We can start by considering a few key aspects of the function, such as its roots, vertex, and behavior. ### 1. Finding the Roots: To find the roots of the function, we set \( f(x) = 0 \): \[x^2 - 2x = 0\] Factoring the equation: \[x(x - 2) = 0\] This gives us the roots: \[x = 0 \quad \text{or} \quad x = 2\] ### 2. Vertex of the Parabola: Since \( f(x) \) is a quadratic function, its graph is a parabola. The vertex form of a quadratic function \( ax^2 + bx + c \) is given by: \[x = -\frac{b}{2a}\] For the function \( f(x) = x^2 - 2x \), we have \( a = 1 \) and \( b = -2 \). Plugging these values into the vertex formula: \[x = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1\] To find the y-coordinate of the vertex, we substitute \( x = 1 \) back into the function: \[f(1) = 1^2 - 2 \cdot 1 = 1 - 2 = -1\] Thus, the vertex of the parabola is at \( (1, -1) \). ### 3. Behavior of the Function: - The parabola opens upwards because the coefficient of \( x^2 \) is positive (\( a = 1 \)). - The vertex represents the minimum point of the function. - The parabola intersects the x-axis at \( x = 0 \) and \( x = 2 \).