Answer: Therefore, the equation of the axis of symmetry is x = 4.
Explanation:
To graph the equation y = -x² + 8x - 12, we can start by finding the vertex and the roots.
1. Vertex: The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a, b, and c are the coefficients of the quadratic equation in the form ax² + bx + c. In this case, a = -1, b = 8, and c = -12. Plugging these values into the formula, we get x = -8 / (2 * -1) = 4.
To find the y-coordinate of the vertex, substitute the x-coordinate (4) back into the equation: y = -(4)² + 8(4) - 12 = -16 + 32 - 12 = 4. Therefore, the vertex is (4, 4).
2. Roots: To find the roots, set y = 0 and solve for x. In this case, we need to solve -x² + 8x - 12 = 0. You can use factoring, completing the square, or the quadratic formula to find the roots. Let's use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
Plugging in the values, we get x = (-(8) ± √((8)² - 4(-1)(-12))) / (2(-1)). Simplifying further, we have x = (-(8) ± √(64 - 48)) / (-2), which becomes x = (-(8) ± √(16)) / (-2). This simplifies to x = (-(8) ± 4) / (-2).
So, the roots are x = (8 + 4) / 2 = 6/2 = 3 and x = (8 - 4) / 2 = 4/2 = 2.
Now, we can plot the points on the graph:
- Vertex: (4, 4)
- Roots: (3, 0) and (2, 0)