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Answer:
A. f(x) = (x -3)² -1
B. vertex: (3, -1); axis of symmetry: x=3; x-intercepts: {2, 4}; y-intercept: 8
C. see attached for a graph
Step-by-step explanation:
A. Put the function in vertex form by completing the square. Start by putting parentheses around the terms containing x.
f(x) = (x² -6x) +8
Now, identify the coefficient of x (-6) and square half of that. (-6/2)² = 9. Add this value inside parentheses, and subtract this value outside parentheses.
f(x) = (x² -6x +9) +8 -9
Rewrite the trinomial in parentheses as a square. The constant in the binomial will be the value you squared in the previous step.
f(x) = (x -3)² -1 . . . . . vertex form; a(x -h)²+k for vertex (h, k)
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B. The vertex and the axis of symmetry can be read from the equation. Here, the vertex is (3, -1). The axis of symmetry is the vertical line through the vertex: x = 3. The x-intercepts are found by solving f(x) = 0.
f(x) = 0
(x -3)² -1 = 0
(x -3)² = 1 . . . . . add the opposite of the 'k' value
x -3 = ±√1 . . . . take the square root. We need both signs.
x = 3 ± 1 . . . . . add 3 to get x by itself. These are the x-intercepts.
x = 3-1 = 2 and x = 3+1 = 4
The x-intercepts are 2 and 4, points (2, 0) and (4, 0).
The y-intercept is the value of f(0). It is the constant in the original form of the function. f(0) = 8 . . . . y-intercept
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C. The graph is attached. The various points above are marked, and the axis of symmetry is shown.
