Simplify the following. (7+5i)^2−(3+2i)=
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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>
