What are the key points on the graph of y = x ^ 2 - 16x + 28% Name the vertex, x- intercepts, and y- intercept . (2 points ) vertex: (smaller) x-intercept: (larger) x-intercept: y -intercept:
To find the vertex, x-intercepts, and y-intercept of the graph y = x^2 - 16x + 28, we can start by using the quadratic formula.
The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a)
Comparing the equation y = x^2 - 16x + 28 to the standard quadratic equation ax^2 + bx + c, we have a = 1, b = -16, and c = 28.
1. Vertex:
The x-coordinate of the vertex can be found using the formula: x = -b / (2a)
x = -(-16) / (2 * 1) = 16 / 2 = 8
To find the y-coordinate of the vertex, substitute the x-coordinate into the equation:
y = (8)^2 - 16(8) + 28 = 64 - 128 + 28 = -36
So the vertex is (8, -36).
2. X-intercepts:
To find the x-intercepts, set y = 0 and solve for x.
0 = x^2 - 16x + 28
Using the quadratic formula:
x = (-(-16) ± √((-16)^2 - 4 * 1 * 28)) / (2 * 1)
x = (16 ± √(256 - 112)) / 2
x = (16 ± √144) / 2
x = (16 ± 12) / 2
x1 = (16 + 12) / 2 = 28 / 2 = 14
x2 = (16 - 12) / 2 = 4 / 2 = 2
So the x-intercepts are 14 and 2.
3. Y-intercept:
To find the y-intercept, set x = 0 and solve for y.
y = (0)^2 - 16(0) + 28 = 0 - 0 + 28 = 28
So the y-intercept is 28.
In conclusion, the key points on the graph of y = x^2 - 16x + 28 are:
Vertex: (8, -36)
X-intercepts: (14, 0) and (2, 0)
Y-intercept: (0, 28)