URGENT WILL GIVE BRAINLIEST if <1=115 find the other missing angles
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Answer:
see explanation
Step-by-step explanation:
(9)
Expand and simplify right side and compare coefficients of like terms on left side.
(x - 2)(x² + ax + b) + c
= x³ + ax² + bx - 2x² - 2ax - 2b + c
= x³ + x²(a - 2) + x(b - 2a) - 2b + c
compare with
x³ + 2x² - 3x + 4
x² terms
a - 2 = 2 ( add 2 to both sides )
a = 4
x terms
b - 2a = - 3 ← substitute a = 4
b - 8 = - 3 ( add 8 to both sides )
b = 5
constant terms
- 2b + c = 4 ← substitute b = 5
- 10 + c = 4 ( add 10 to both sides )
c = 14
Thus a = 4, b = 5 and c = 14
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(10)
Since both functions cross the x- axis at - 2 then (- 2, 0) satisfies both, that is
f(- 2) = + a(- 2)³ + b(- 2)² + 36(- 2) + 144 = 0, that is
- 16 - 8a + 4b - 72 + 144 = 0
- 8a + 4b + 56 = 0
- 8a + 4b = - 56 → (1)
and
g(- 2) = + (a + 3)(- 2)³ - 23(- 2)² + (b + 10)(- 2) + 40 = 0, that is
- 16 - 8(a + 3) - 92 - 2(b + 10) + 40 = 0
- 16 - 8a - 24 - 92 - 2b - 20 + 40 = 0
- 8a - 2b - 112 = 0
- 8a - 2b = 112 → (2)
Subtract (1) from (2) term by term
- 6b = 168 ( divide both sides by - 6 )
b = - 28
Substitute b = - 28 into (2)
- 8a + 56 = 112 ( subtract 56 from both sides )
- 8a = 56 ( divide both sides by - 8 )
a = - 7
Thus a = - 7 and b = - 28
The two graphs are represented below.
Answer and Step-by-step explanation: One graph can "transform" into another through changes in the function.
There are 3 ways to change a function:
- <u>Shifting</u>: it adds or subtracts a constant to one of the coordinates, thus changing the graph's location. When the <em><u>y-coordinate</u></em> is<em> </em>added or subtract and the x-coordinate is unchanged, there is a <em><u>vertical</u></em> <u><em>shift</em></u>. If it is the <em><u>x-coordinate</u></em> which changes and y-coordinate is kept the same, the shift is a <em><u>horizontal</u></em> <u><em>shift</em></u>;
- <u>Scaling</u>: it multiplies or divides one of the coordinates by a constant, thus changing position and appearance of the graph. If the <em>y-coordinate</em> is multiplied or divided by a constant but x-coordinate is the same, it is a <em>vertical scaling</em>. If the <em>x-coordinate</em> is changed by a constant and y-coordinate is not, it is a <em>horizontal</em> <em>scaling</em>;
- <u>Reflecting</u>: it's a special case of scaling, where you can multiply a coordinate per its opposite one;
Now, the points for f(x) are:
(-5,0) (0,6) (5,-4) (8,0)
And the points for g(x) are:
(-5,-3) (0,-9) (5,1) (8,-3)
Comparing points:
(-5,0) → (-5,-3)
(0,6) → (0,-9)
(5,-4) → (5,1)
(8,0) → (8,-3)
It can be noted that x-coordinate is kept the same; only y-coordinate is changing so we have a vertical change. Observing the points:
(-5,0-3) → (-5,-3)
(0,6-15) → (0,-9)
(5,-4+5) → (5,1)
(8,0-3) → (8,-3)
Then, the vertical change is a <u>Vertical</u> <u>Shift</u>.
Another observation is that y-coordinate of f(x) is the opposite of g(x). for example: At the second point, y-coordinate of f(x) is 6, while of g(x) is -9. So, this transformation is also a <u>Reflection</u>.
<u>Range</u> <u>of</u> <u>a</u> <u>function</u> is all the values y can assume after substituting the x-values.
<u>Domain</u> <u>of</u> <u>a</u> <u>function</u> is all the values x can assume.
Reflection doesn't change range nor domain of a function. However, vertical or horizontal translations do.
Any vertical translation will change the range of a function and keep domain intact.
Then, for f(x) and g(x):
graph translation domain range
f(x) none [-5,8] [-4,6]
g(x) vertical shift [-5,8] [-9,1]
<u>In conclusion, this transformation (or translation) will affect the range of g(x)</u>
