Using translation concepts, it is found that:
a.
- The equation is: y = -1.5(x + 4)² + 9.
- The domain is all real values.
b.
- The rule is [tex]y = -\sqrt{x + 5} + 10[/tex].
What is a translation?
A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction either in it’s definition or in it’s domain. Examples are shift left/right or bottom/up, vertical or horizontal stretching or compression, and reflections over the x-axis or the y-axis.
Quadratic function
The quadratic function has vertex at (-4,-6), hence:
y = a(x + 4)² - 6
When x = -6, y = 0, hence we use it to find a as follows:
0 = 4a - 6
a = 1.5.
Hence the function is:
y = 1.5(x + 4)² - 6.
When it is reflected over the x-axis, we have that:
y -> -y.
Hence:
y = -1.5(x + 4)² + 6.
When it is moved up 3 units, we have that y -> y + 3, hence:
y = -1.5(x + 4)² + 9.
The function is defined for all real values, assuming values from negative infinity to 9, hence:
- The equation is: y = -1.5(x + 4)² + 9.
- The domain is all real values.
Square root function
The parent square root function is given by:
[tex]y = \sqrt{x}[/tex]
From the graph, it is found that:
- The function was reflected over the x-axis, hence y -> -y.
- The function was shifted one unit to right, hence x -> x - 1.
- The function was shifted two units up, hence y -> y + 2.
Hence the rule is:
[tex]y = -\sqrt{x - 1} + 2[/tex]
The translations given by the item are:
- 6 units left, hence x -> x + 6.
- 8 units up, hence y -> y + 8.
Thus:
[tex]y = -\sqrt{x + 5} + 10[/tex]
Then the function will be defined for values of -5 and greater, assuming values of 10 and less, hence:
- The rule is [tex]y = -\sqrt{x + 5} + 10[/tex].
More can be learned about translation concepts at https://brainly.com/question/4521517
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