Certainly! Here’s a narrative explanation for the process of graphing the quadratic function **y = x² - 6x + 8**, including how to find the vertex, the axis of symmetry, and the x-intercepts: --- To graph the quadratic function y = x² - 6x + 8, we start by recognizing that this is a parabola that opens upwards because the coefficient of x² is positive. The general form of a quadratic equation is y = ax² + bx + c, where in this case, a = 1, b = -6, and c = 8. **1. Finding the Vertex:** The vertex of a parabola in standard form can be found using the formula x = -b/(2a). Plugging in our values: x = -(-6) / (2 * 1) x = 6 / 2 x = 3 To find the y-coordinate of the vertex, substitute x = 3 back into the original equation: y = (3)² - 6(3) + 8 y = 9 - 18 + 8 y = -1 So, the vertex is at (3, -1). **2. Axis of Symmetry:** The axis of symmetry is a vertical line that passes through the vertex. For this function, it is x = 3. **3. Finding the x-intercepts:** The x-intercepts are the points where y = 0. So, set the equation to zero and solve for x: 0 = x² - 6x + 8 This quadratic can be factored: 0 = (x - 4)(x - 2) Setting each factor to zero gives us the solutions: x - 4 = 0 → x = 4 x - 2 = 0 → x = 2 So, the x-intercepts are at (4, 0) and (2, 0). **4. Graphing the Function:** - The vertex (3, -1) is the lowest point since the parabola opens upward. - The axis of symmetry is the vertical line x = 3. - The parabola crosses the x-axis at (2, 0) and (4, 0). - The y-intercept can also be found by setting x = 0: y = 0² - 6*0 + 8 = 8, so the y-intercept is at (0, 8). **Summary:** When graphing, plot the vertex at (3, -1), the axis of symmetry at x = 3, the x-intercepts at (2, 0) and (4, 0), and the y-intercept at (0, 8). Draw a smooth curve through these points to complete the graph of the parabola. --- This process walks you step-by-step through understanding and graphing the quadratic function y = x² - 6x + 8.