Jake has been in band twice as many years as billy. Together they have been in band a total of 18 years. How long has billy been
1 answer:
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Answer:
- turning point: (0, -1)
- domain, range: all real numbers
- x-intercept: (1/27, 0)
- y-intercept: (0, -1)
- transformations: vertical expansion by a factor of 3; translation down 1
Step-by-step explanation:
There are a couple of transformations that may be of interest:
g(x) = k·f(x) . . . . vertical scaling by a factor of k
g(x) = f(x) +k . . . vertical translation by k units (up)
g(x) = f(x -k) . . . horizontal translation by k units (right); <em>not used here</em>
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Unlike the square root function, which is undefined for negative values, the cube root function is defined for all real numbers. Its domain and range are all real numbers.
The turning point of a cube-root function is the origin. Here, that has been translated down 1 unit, so it is (0, -1). That is also the y-intercept.
The x-intercept is the value of x where g(x)=0:
0 = 3∛x -1
1 = 3∛x
1/3 = ∛x
(1/3)³ = x = 1/27
The x-intercept is (1/27, 0).
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<u>Transformations</u>
As we discussed above, the addition of -1 to the parent function causes it to be translated down 1 unit.
The multiplication of the parent function by 3 causes it to be vertically expanded by a factor of 3.
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Explanation:
<u>Given</u>:
- The attached figure showing circle O, chord BC, central angle BOC and inscribed angle BAC
- angle BAC = α + β
<u>Prove</u>:
- angle BOC = 2×angle BAC
<u>Proof</u>:
∠BOA +∠BOC +∠AOC = 360° . . . . . sum of arcs of a circle is 360°
2α +∠BOA = 180°, 2β +∠AOC = 180° . . . . . sum of triangle angles is 180°
∠BOA = 180° -2α, ∠AOC = 180° -2β . . . . solve statement 2 for central angles
(180° -2α) +∠BOC +(180° -2β) = 360° . . . . . substitute into statement 1
∠BOC = 2(α +β) . . . . . add 2α+2β-360° to both sides
∠BOC = 2×∠BAC . . . . . substitute given for α+β; the desired conclusion
