Answer:
1) a = 1, b = -8, c = 17; Vertex: (4, 1)
2) a = -1, b = -2. c = -2; Vertex: (-1, -1)
3) a = -1, b = 6, c = -8; Vertex: (3, 1)
4) a = -3, b = 6, c = 0; Vertex: (1, 3)
5) a = -2, b = -16, c = -31; Vertex: (-4, 1)
6) a = -1/2 or -0.5, b = -4, c = -6; Vertex: (-4, 2)
Explanation:
The quadratic functions listed are all in standard form:
y = ax² + bx + c
where a, b, and c, are coefficients for each of the terms.
Vertex
To find the vertex of a parabolic equation in standard form. Calculate -b/2a. This will be your x-coordinate. Then substitute this back into f(x) to obtain the y-coordinate; The calculated point is your vertex.
1) x = - b / 2a = - (-8) / 2 (1) = 8 / 2 = 4
f(4) = 4² - 8 (4) + 17 = 16 - 32 + 17 = 1
Vertex: (4, 1)
2) x = -b / 2a = - (-2) / 2 (-1) = 2 / (-2) = -1
f(-1) = - (-1)² - 2 (-1) - 2 = -1 + 2 - 2 = -1
Vertex: (-1, -1)
3) x = - b / 2a = - (6) / 2 (-1) = -6 / -2 = 3
f(3) = - (3)² + 6 (3) -8 = -9 + 18 - 8 = 1
Vertex: (3, 1)
4) x = - b / 2a = - (6) / 2 (-3) = -6 / -6 = 1
f(1) = -3 (1)² + 6 (1) = -3 + 6 = 3
Vertex: (1, 3)
5) x = - b / 2a = - (-16) / 2(-2) = 16 / -4 = -4
f(-4) = -2 (-4)² - 16 (-4) - 31 = -32 + 64 - 31 = 1
Vertex: (-4, 1)
6) x = - b / 2a = - (-4) / 2 (-0.5) = 4 / -1 = -4
f (-4) = (-0.5) (-4)² - 4 (-4) - 6 = -8 + 16 - 6 = 2
Vertex: (-4, 2)