Which statement describes the translation of y = −one half (x − 2)2 − 2 from standard position?
1 answer:
A quadratic in vertex form can be represented as
a represents reflection over the x-axis, and a vertical stretch or compress
- is reflection and a fraction (1/2) represents a compression.
-h represents a shift of that many units to the right (-2 shifts to the right two units)
k represents a shift up or down (-2 is shifting down 2 units_
Reflected over the x-axis, Vertically compressed by a factor of 1/2, shifted 2 units to the right, and shifted 2 units down
a represents reflection over the x-axis, and a vertical stretch or compress
- is reflection and a fraction (1/2) represents a compression.
-h represents a shift of that many units to the right (-2 shifts to the right two units)
k represents a shift up or down (-2 is shifting down 2 units_
Reflected over the x-axis, Vertically compressed by a factor of 1/2, shifted 2 units to the right, and shifted 2 units down
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Note that f(x) as given is <em>not</em> invertible. By definition of inverse function,
which is a cubic polynomial in with three distinct roots, so we could have three possible inverses, each valid over a subset of the domain of f(x).
Choose one of these inverses by restricting the domain of f(x) accordingly. Since a polynomial is monotonic between its extrema, we can determine where f(x) has its critical/turning points, then split the real line at these points.
f'(x) = 3x² - 1 = 0 ⇒ x = ±1/√3
So, we have three subsets over which f(x) can be considered invertible.
• (-∞, -1/√3)
• (-1/√3, 1/√3)
• (1/√3, ∞)
By the inverse function theorem,
where f(a) = b.
Solve f(x) = 2 for x :
x³ - x + 2 = 2
x³ - x = 0
x (x² - 1) = 0
x (x - 1) (x + 1) = 0
x = 0 or x = 1 or x = -1
Then can be one of
• 1/f'(-1) = 1/2, if we restrict to (-∞, -1/√3);
• 1/f'(0) = -1, if we restrict to (-1/√3, 1/√3); or
• 1/f'(1) = 1/2, if we restrict to (1/√3, ∞)