<em><u>Correct steps are shown</u></em>
<em><u>Correct simplified expression is:</u></em>
<em><u>Solution:</u></em>
<em><u>Given that: Mara simplifies the expression</u></em>
<em><u>Correct steps are:</u></em>
From given,
<em><u>Use the distributive property,</u></em>
a(b + c) = ab + bc
Therefore,
Group the like terms
Thus the expression is simplified
Answer:
Step-by-step explanation:
Given
See attachment
From the attachment, we have:
First, we need to calculate length LM,
Using Pythagoras theorem:
Collect Like Terms
Solving (a):
Substitute values for MN and LN
Solving (b):
Substitute values for LM and MN
Solving (c):
Substitute values for LN and LM
Solving (d):
Substitute values for LM and LN
The vertex form of g(x) is ...
... g(x) = -(x -3)² +4
This is offset to the right by 3 and up by 4 from the parent function. (It is also first reflected across the x-axis.)
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<em>Vertex form</em>
You know the leading coefficient is -1 because that's what it is for x² in the given form. When you factor -1 from the first two terms, of the given form, you have ...
... g(x) = -1(x² -6x) -5
Half the x coefficient inside parentheses will be the constant in the squared binomial term, so that term is (x -3)². The constant in that square is +9, so adding that value inside and outside parentheses in g(x) gives ...
... g(x) = -1(x² -6x +9) -5 +9
... g(x) = -(x -3)² +4 . . . . . vertex form
_____
<em>About transformations</em>
g(x) = f(x -a) causes the graph of f(x) to be shifted "a" units to the right. For a function f(x) with an axis of symmetry at x=0, it moves the axis of symmetry to x=a.
g(x) = f(x) +a causes the graph of f(x) to be shifted "a" units up.
g(x) = -f(x) causes the graph of f(x) to be reflected across the x-axis.
Here, we have all three of these transformations. First is the reflection:
... f₁(x) = -f(x) = -x²
Then we have shifting to the right 3 units. (also moves the axis of symmetry)
... f₂(x) = f₁(x-3) = -(x -3)²
Finally, we have shifting up 4 units.
... g(x) = f₂(x) +4 = -(x -3)² +4
