Question

f(x)=x^2-2x

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Santino

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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 \( x = 2 \) .

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