13) \( y=x^{2}-6 x+6 \)

Sure! Let's analyze the quadratic function: \[ y = x^{2} - 6x + 6 \] ### 1. **Standard Form and Coefficients** The given equation is in the standard quadratic form: \[ y = ax^{2} + bx + c \] Where: - \( a = 1 \) - \( b = -6 \) - \( c = 6 \) ### 2. **Vertex of the Parabola** The vertex of a parabola defined by a quadratic function can be found using the formula: \[ x = -\frac{b}{2a} \] Substituting the values: \[ x = -\frac{-6}{2 \times 1} = 3 \] Now, substitute \( x = 3 \) back into the original equation to find the \( y \)-coordinate of the vertex: \[ y = (3)^2 - 6(3) + 6 = 9 - 18 + 6 = -3 \] **Vertex:** \( (3, -3) \) ### 3. **Axis of Symmetry** The axis of symmetry is a vertical line that passes through the vertex: \[ x = 3 \] ### 4. **Y-intercept** The y-intercept occurs where \( x = 0 \): \[ y = (0)^2 - 6(0) + 6 = 6 \] **Y-intercept:** \( (0, 6) \) ### 5. **X-intercepts (Roots)** To find the x-intercepts, set \( y = 0 \) and solve for \( x \): \[ 0 = x^{2} - 6x + 6 \] Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substituting the coefficients: \[ x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(6)}}{2(1)} \] \[ x = \frac{6 \pm \sqrt{36 - 24}}{2} \] \[ x = \frac{6 \pm \sqrt{12}}{2} \] \[ x = \frac{6 \pm 2\sqrt{3}}{2} \] \[ x = 3 \pm \sqrt{3} \] **X-intercepts:** \( x = 3 + \sqrt{3} \) and \( x = 3 - \sqrt{3} \) ### 6. **Direction of the Parabola** Since the coefficient \( a = 1 \) is positive, the parabola opens **upwards**. ### 7. **Graphing the Parabola** - **Vertex:** \( (3, -3) \) - **Axis of Symmetry:** \( x = 3 \) - **Y-intercept:** \( (0, 6) \) - **X-intercepts:** \( (3 + \sqrt{3}, 0) \) and \( (3 - \sqrt{3}, 0) \) **Sketch Overview:** 1. Plot the vertex at \( (3, -3) \). 2. Draw the axis of symmetry as a dashed line \( x = 3 \). 3. Plot the y-intercept at \( (0, 6) \). 4. Plot the x-intercepts at \( (3 + \sqrt{3}, 0) \) and \( (3 - \sqrt{3}, 0) \). 5. Draw a symmetric U-shaped curve opening upwards through these points. ### 8. **Summary of Key Points** - **Vertex:** \( (3, -3) \) - **Axis of Symmetry:** \( x = 3 \) - **Y-intercept:** \( y = 6 \) - **X-intercepts:** \( x = 3 \pm \sqrt{3} \) - **Parabola Direction:** Opens upwards If you have a specific aspect of the quadratic function you'd like to explore further (e.g., completing the square, optimization, real-world applications), feel free to ask!