Answer:
<u>1. Type of function</u>: absolute value function
<u>2. The three transformations from the parent function f(x) = |x| are</u>:
- Translation 3 units to the left
- Vertical stretch with a scale factor of 2
- Translation 5 units downward
Explanation:
<u>1. Type of function</u>
The parent function |x| is the absolute value function. It returns the positive value of the argument (x).
Thus the function f(x) = 2|x + 3| - 5 is also an absolute value function.
It returns the positive value of x + 3, then multiplyes it by 2, and finally subtract 5.
This is a piecewise function.
For the values of x ≥ - 3, the output is 2(x +3) - 5 = 2x + 6 - 5 = 2x + 1.
For the values of x < - 3, the output is 2 (-x - 3) - 5 = -2x - 6 - 5 = -2x - 11.
The vertex of this function is at x = - 3: 2 (- 3 + 3) - 5 = 2(0) - 5 = 0 - 5 = 5.
Thus the vertex is (-3, 5).
<u>2. Transformations from the parent function f(x) = |x|</u>.
When you know the parent function and the daughter function you can know the transformations done of the former to get the later by some simple rules.
a)<u> Translation in the horizontal direction</u>.
When you add a positive value to the argument the function is translated to the left.
Thus, when you add 3 to x, |x| becomes |x + 3| and it is a translation 3 units to the left.
b) <u>Vertical stretch</u>
When you multiply the argument by a constant, the function stretches vertically.
Thus, when you multiply |x + 3| by 2, to get 2|x + 3|, the function is vertically stretched by a scale factor of 2.
c) <u>Vertical translation</u>
When you subtract a constant value from the function, you translate it downward.
Thus, when you subtract 5 from 2|x + 3| - 5, you translate the function 5 units downward.
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