Answer: f(x) = 1 + √(- x² - 4x + 21), D: [-7, 3], R: [1, 6]
Explanation:
(x + 2)² + (y - 1)² = 25 ⇒ center (h, k) = (-2, 1) and radius (r) = 5
Equation of a semicircle is to solve the equation for y.
(x + 2)² + (y - 1)² = 25
(y - 1)² = 25 - (x + 2)²
(y - 1)² = 25 - (x² + 4x + 4)
(y - 1)² = 25 - x² - 4x - 4
(y - 1)² = - x² - 4x + 21
√(y - 1)² = √(- x² - 4x + 21)
y - 1 = +/- √(- x² - 4x + 21)
y = 1 +/- √(- x² - 4x + 21)
Since we are looking for the top half, the equation is:
y = 1 + √(- x² - 4x + 21)
in function format:
f(x) = 1 + √(- x² - 4x + 21)
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NOTE: The easiest way to find the domain and range is to graph the circle. Below is how to find the domain and range algebraically.
Domain: square root must be greater than or equal to zero
- x² - 4x + 21 ≥ 0
x² + 4x - 21 ≤ 0 divided both sides by -1
(x + 7)(x - 3) = 0 factored to find the zeros
x + 7 = 0 x - 3 = 0
x = -7 x = 3
The domain must be between -7 and 3 inclusively ; -7 ≤ x ≤ 3.
Interval Notation: D: [-7, 3]
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Range: find the y-value at one of the zeros (x = -7 or x = 3) and the y-value of the vertex (the axis of symmetry goes through the center, x = -2)
f(x) = 1 + √(- x² - 4x + 21)
f(-7) = 1 + √(- (-7)² - 4(-7) + 21)
= 1 + √(-49 + 28 + 21)
= 1 + √(0)
= 1
f(-2) = 1 + √(- (-2)² - 4(-2) + 21)
= 1 + √(-4 +8 + 21)
= 1 + √(25)
= 1 + 5
= 6
the range must be between 1 and 6 inclusively ; 1 ≤ y ≤ 6
Interval Notation: R: [1, 6]
