Graph the equation y, equals, minus, x, squared, plus, 6, x, minus, 8y=−x 2 +6x−8 on the accompanying set of axes. You must plot 5 points including the roots and the vertex.

Sure, let's graph the equation y = -x^2 + 6x - 8.

Step 1: Identify the roots of the equation. The roots are the x-values where y = 0. We can find these by setting the equation equal to zero and solving for x:

0 = -x^2 + 6x - 8

This is a quadratic equation, and we can solve it using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / (2a). In this case, a = -1, b = 6, and c = -8.

x = [-6 ± sqrt((6)^2 - 4*(-1)(-8))] / 2(-1) x = [-6 ± sqrt(36 - 32)] / -2 x = [-6 ± sqrt(4)] / -2 x = [-6 ± 2] / -2 x = 4, 2

So, the roots of the equation are x = 2 and x = 4. These are the first two points we will plot: (2, 0) and (4, 0).

Step 2: Find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula -b / (2a), and the y-coordinate can be found by substituting this x-value back into the equation.

x = -6 / (2*-1) = 3 y = -(3)^2 + 6*3 - 8 = -9 + 18 - 8 = 1

So, the vertex of the parabola is (3, 1). This is the third point we will plot.

Step 3: Choose two more points to plot. Since we know the parabola is symmetric, we can choose x-values that are equidistant from the vertex. Let's choose x = 1 and x = 5.

y = -(1)^2 + 61 - 8 = -1 + 6 - 8 = -3 y = -(5)^2 + 65 - 8 = -25 + 30 - 8 = -3

So, the last two points we will plot are (1, -3) and (5, -3).

Step 4: Plot the five points on the graph: (2, 0), (4, 0), (3, 1), (1, -3), and (5, -3). Draw a smooth curve through the points to complete the graph of the equation.