Graph the quadratic function below \( y=-2 x^{2}+4 x-2 \)

1. **Write the equation and factor the coefficient of \( x^2 \):** The quadratic function is \[ y = -2x^2 + 4x - 2. \] Factor out \(-2\) from the quadratic and linear terms: \[ y = -2\left(x^2 - 2x\right) - 2. \] 2. **Complete the square:** To complete the square for \( x^2 - 2x \), take half the coefficient of \( x \) (which is \(-2\)) and square it: \[ \left(\frac{-2}{2}\right)^2 = 1. \] Add and subtract \(1\) inside the parentheses: \[ y = -2\left(x^2 - 2x + 1 - 1\right) - 2. \] Thus, \[ y = -2\left[(x-1)^2 - 1\right] - 2. \] 3. **Rewrite in vertex form:** Distribute the \(-2\): \[ y = -2(x-1)^2 + 2 - 2. \] Combine the constants: \[ y = -2(x-1)^2. \] The vertex form is \[ y = -2(x-1)^2. \] 4. **Identify key features:** - **Vertex:** The vertex of a quadratic in the form \( y = a(x-h)^2 + k \) is \((h, k)\). Here, the vertex is \[ (1, 0). \] - **Axis of symmetry:** This is the vertical line \( x = h \), so \[ x = 1. \] - **Direction of opening:** Since the coefficient \( a = -2 \) is negative, the parabola opens downward. - **\( y \)-intercept:** Substitute \( x = 0 \) into the original equation: \[ y = -2(0)^2 + 4(0) - 2 = -2, \] so the \( y \)-intercept is \[ (0, -2). \] - **\( x \)-intercepts:** Set \( y = 0 \): \[ 0 = -2(x-1)^2. \] Divide by \(-2\): \[ 0 = (x-1)^2. \] Take the square root: \[ x - 1 = 0 \quad \Rightarrow \quad x = 1. \] So there is a single \( x \)-intercept at \[ (1, 0), \] which is also the vertex. 5. **Sketching the graph:** - Plot the vertex at \((1, 0)\). - Draw the axis of symmetry as the vertical line \( x = 1 \). - Mark the \( y \)-intercept at \((0, -2)\). - Note that the parabola touches the \( x \)-axis only at the vertex \((1, 0)\) and opens downward. - You can choose additional points for a smoother curve; for example, for \( x = 2 \): \[ y = -2(2-1)^2 = -2(1)^2 = -2, \] which gives the point \((2, -2)\). Its symmetric point about \( x=1 \) is \((0, -2)\) (which is already plotted as the \( y \)-intercept). 6. **Conclusion:** The graph of \[ y = -2x^2 + 4x - 2 \] is a downward opening parabola with vertex \((1, 0)\), a single \( x \)-intercept at \((1, 0)\), and a \( y \)-intercept at \((0, -2)\). The key features are summarized as: - **Vertex:** \( (1, 0) \) - **Axis of symmetry:** \( x = 1 \) - **Opens:** Downward (since \( a < 0 \)) - **\( y \)-intercept:** \( (0, -2) \) - **\( x \)-intercept:** \( (1, 0) \) This completes the step-by-step solution for graphing the quadratic function.