Graph the equation y, equals, minus, x, squared, plus, 12, x, minus, 35y=−x 2 +12x−35 on the accompanying set of axes. You must plot 5 points including the roots and the vertex.

Step 1: Identify the equation type The given equation is a quadratic equation in the form y = ax^2 + bx + c, where a = -1, b = 12, and c = -35.

Step 2: Find the roots The roots of the equation can be found using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / 2a. Substituting the values of a, b, and c, we get: x = [-12 ± sqrt((12)^2 - 4*(-1)(-35))] / 2(-1) x = [-12 ± sqrt(144 - 140)] / -2 x = [-12 ± sqrt(4)] / -2 So, the roots are x = 5 and x = 7. These give us two points on the graph: (5,0) and (7,0).

Step 3: Find the vertex The vertex of a parabola y = ax^2 + bx + c is given by the point (h, k), where h = -b/2a and k = c - b^2/4a. Substituting the values of a and b, we get: h = -b/2a = -12 / 2*(-1) = 6 k = c - b^2/4a = -35 - (12)^2 / 4*(-1) = -35 - 36 = -71 So, the vertex is at the point (6, -71).

Step 4: Choose two more points To get a better idea of the shape of the graph, let's choose two more points. We can choose x = 0 and x = 8 for simplicity. Substituting these values into the equation, we get: For x = 0, y = -(0)^2 + 120 - 35 = -35. So one point is (0, -35). For x = 8, y = -(8)^2 + 128 - 35 = -64 + 96 - 35 = -3. So another point is (8, -3).

Step 5: Plot the points and draw the graph Now we can plot the five points we found: (5,0), (7,0), (6,-71), (0,-35), and (8,-3). Connect the points with a smooth curve to complete the graph of the equation y = -x^2 + 12x - 35. The graph opens downwards because the coefficient of x^2 is negative.