Graph the equation \( y=-x^{2}+14 x-48 \) on the accompanying including the roots and the vertex.

To graph the equation \( y = -x^2 + 14x - 48 \), we first need to identify the roots and the vertex of the parabola. ### Step 1: Find the Roots To find the roots, we can set \( y = 0 \): \[ -x^2 + 14x - 48 = 0 \] Multiplying through by -1 gives: \[ x^2 - 14x + 48 = 0 \] Next, we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = -14 \), and \( c = 48 \). Calculating the discriminant: \[ b^2 - 4ac = (-14)^2 - 4 \cdot 1 \cdot 48 = 196 - 192 = 4 \] Now, substituting into the quadratic formula: \[ x = \frac{14 \pm \sqrt{4}}{2 \cdot 1} = \frac{14 \pm 2}{2} \] This gives us two roots: \[ x = \frac{16}{2} = 8 \quad \text{and} \quad x = \frac{12}{2} = 6 \] So the roots are \( x = 6 \) and \( x = 8 \). ### Step 2: Find the Vertex The vertex of a parabola given by the equation \( y = ax^2 + bx + c \) can be found using the formula: \[ x = -\frac{b}{2a} \] In our case, \( a = -1 \) and \( b = 14 \): \[ x = -\frac{14}{2 \cdot -1} = 7 \] Now, we can find the corresponding \( y \)-value by substituting \( x = 7 \) back into the original equation: \[ y = -7^2 + 14 \cdot 7 - 48 \] \[ y = -49 + 98 - 48 \] \[ y = 1 \] So the vertex is at the point \( (7, 1) \). ### Step 3: Graph the Parabola Now we can summarize the key points for the graph: - **Roots**: \( (6, 0) \) and \( (8, 0) \) - **Vertex**: \( (7, 1) \) The parabola opens downwards (since the coefficient of \( x^2 \) is negative). ### Step 4: Sketch the Graph 1. Plot the points \( (6, 0) \), \( (7, 1) \), and \( (8, 0) \). 2. Draw a smooth curve through these points, making sure it opens downwards. The graph will look like a downward-opening parabola with the vertex at \( (7, 1) \) and crossing the x-axis at \( (6, 0) \) and \( (8, 0) \). ### Final Graph You can use graphing software or graph paper to accurately plot these points and draw the parabola.