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a) y = (x -4)² -4
b) (4, -4)
c) translated 4 right, 4 down
Step-by-step explanation:
a) Apparently "graphing form" is a reference to "vertex form." That form is ...
y = a(x -h)² +k . . . . . . parabola with vertex (h, k) and vertical scale factor 'a'
The coefficient of x² in your given equation is 1, so a=1. The values of h and k can be found as follows.
'h' is the opposite of half the x-coefficient in your equation, so is h = -(-8/2) = 4. The value of k is the square of h, subtracted from the constant in your equation: k = 12 -h² = 12 -16 = -4. (This applies when a=1, as here.) These calculations are the "cut to the chase" version of "completing the square."
The method of "completing the square" has you consider the x-terms separately from the constant term:
y = (x² -8x) +12
To "complete the square", you want to make the terms in parentheses be a perfect square trinomial. You do that by adding the square of half the x-coefficient. In order to keep the same equation, you need to subtract an equivalent amount outside parentheses:
y = (x² -8x +(-8/2)²) +12 -(-8/2)² . . . . . add and subtract (-8/2)²
y = (x -4)² +(12 -16)
Then the vertex form is ...
y = (x -4)² -4 . . . . and the vertex is (h, k) = (4, -4).
The graphing calculator plot attached confirms this vertex value.
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The problem statement tells you that you can also find this form by averaging the x-intercepts. If you factor your equation, you get ...
y = (x -6)(x -2)
The x-values that make these factors zero {6, 2} are the x-intercepts. Their average (6+2)/2 = 4 is the x-coordinate of the vertex (h). You can find the y-coordinate of the vertex by evaluating y when x=4: y = 4² -8·4 +12 = -4. So, the vertex by this method is (h, k) = (4, -4), and the equation in graphing form is ...
y = (x -4)² -4 . . . . as above.
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b) As we saw above, the vertex coordinates are (h, k) = (4, -4).
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c) The vertical scale factor 'a' is 1, so the only transformation done on the graph is to move the parent vertex (0, 0) to the location (4, -4). That is, the graph has been translated 4 units right and 4 units down. (In the attached, the parent function is shown with a dashed line.)
