what are the key points on the graph of y = x^2 - 16x + 48? name the vertex, x-intercepts, and y-intercept
To find the key points on the graph of y = x^2 - 16x + 48, let's analyze each point separately:
1. Vertex:
To find the vertex, first identify the coefficient of x^2, which is 1. The x-coordinate of the vertex is given by x = -b/2a, where a is the coefficient of x^2 and b is the coefficient of x. In this case, a = 1 and b = -16. Plugging the values into the formula, we get x = -(-16)/(2*1) = 16/2 = 8. To find the corresponding y-coordinate, we substitute x = 8 into the equation. Thus, y = (8)^2 - 16(8) + 48 = 64 - 128 + 48 = -16. Therefore, the vertex is (8, -16).
2. X-intercepts:
To find the x-intercepts, we set y = 0 and solve for x. So, 0 = x^2 - 16x + 48. Factoring the quadratic equation, we get (x - 4)(x - 12) = 0. Setting each factor equal to zero, we find x - 4 = 0 and x - 12 = 0. Solving these equations, we get x = 4 and x = 12. Therefore, the x-intercepts are (4, 0) and (12, 0).
3. Y-intercept:
To find the y-intercept, we set x = 0 and solve for y. Substituting x = 0 into the equation, we get y = (0)^2 - 16(0) + 48 = 0 - 0 + 48 = 48. Therefore, the y-intercept is (0, 48).
In summary, the key points on the graph of y = x^2 - 16x + 48 are:
- Vertex: (8, -16)
- X-intercepts: (4, 0) and (12, 0)
- Y-intercept: (0, 48)