Y=|-x^2+6x-8|

To find the graph of the equation, let's break it down into two separate cases:

Case 1: x^2 - 6x + 8 ≥ 0

For this case, we have a parabolic curve with a concave upward shape. To find the x-intercepts, we set the equation equal to zero:

x^2 - 6x + 8 = 0

Using the quadratic formula, we find the two x-intercepts:

x = (6 ± √(6^2 - 4(1)(8))) / (2(1)) x = (6 ± √(36 - 32)) / 2 x = (6 ± √4) / 2 x = (6 ± 2) / 2

Simplifying:

x = 4 or x = 2

So the x-intercepts are x = 4 and x = 2.

Now, let's find the vertex of the parabola. The x-value of the vertex can be found using the formula:

x = -b / (2a)

In this case, a = 1 and b = -6, so the x-coordinate of the vertex is:

x = -(-6) / (2(1)) x = 6 / 2 x = 3

To find the y-coordinate of the vertex, we substitute x = 3 into the equation:

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

Thus, the vertex is located at (3, 1).

Case 2: x^2 - 6x + 8 < 0

To solve this inequality, we need to find the x-values that make the expression negative. Setting up a number line, we can see that the inequality is satisfied between the x-intercepts of the parabola found earlier: x = 4 and x = 2.

Now, let's plot the points and draw the graph:

The parabolic curve opens upward through the points (2, 0), (4, 0), and reaches its vertex at (3, 1).

To find the graph of y = |-x^2 + 6x - 8|, we need to reflect the portion of the graph below the x-axis to create the absolute value function.

Thus, the graph will look like the curve of y = x^2 - 6x + 8 below the x-axis and the curve of y = x^2 - 6x + 8 above the x-axis.