Question 207353: Compare the parabola defined by each equation with the standard parabola defined by the equation y = x^2. Describe the corresponding transformations, and include the position of the vertex and the equation of the axis os symmetry.
a.) y = 3x^2 - 8
b.) y = (x - 6)^2 + 4
c.) y = -4(x + 3)^2 - 7
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website! Compare the parabola defined by each equation with the standard parabola defined by the equation y = x^2. Describe the corresponding transformations, and include the position of the vertex and the equation of the axis os symmetry.
Order of transformations:
1. Replace x by -x (reflects graph across y-axis)
2. Replace x by x±k (shifts graph k units horizontally,
left if x-k, and right if x+k)
3. Multiply right side by a positive number k
Stretches vertically by factor of k if k>1
Shrinks vertically by a factor of k if k<1
4. Multiply right side by -1 (reflects graph across the x-axis)
5. Add to or subtract from the right side (shifts graph up if
adding a positive number, and down if adding a negative.)
a.) y = 3x^2 - 8
transformation effect on graph:
y = x^2 <--- original equation, has vertex (0,0)
axis of symmetry x=0
Multiplying right side by 3, which is >1,
y = 3x^2 <--- stretches vertically by factor of 3, still
has vertex (0,0), axis of symmetry x=0
Subtracting 8 from right side
y = 3x^2 - 8 <--- shift 8 units downward, vertex (0,-9)
axis of symmetry x=0
===========================================================
b.) y = (x - 6)^2 + 4
transformation effect on graph:
y = x^2 <--- original equation, has vertex (0,0)
axis of symmetry x=0
Replacing x by (x-6)
y = (x-6)^2 <--- shifts 6 units right, has vertex (6,0)
axis of symmetry x=6
Adding 4 to right side
y = (x-6)^2 + 4 <--- shifts 4 units upward, vertex (6,4)
axis of symmetry x=6
============================================
c.) y = -4(x + 3)^2 - 7
transformation effect on graph:
y = x^2 <--- original equation, has vertex (0,0)
axis of symmetry x=0
Replacing x by (x+3)
y = (x+3)^2 <--- shifts 3 units left, has vertex (-3,0)
axis of symmetry x=-3
Multiplying right side by 4, which is >1,
y = 4(x+3)^2 <--- stretches vertically by factor of 4, still
has vertex (-3,0), axis of symmetry x=-3
Multiplying right side by -1
y = -4(x+3)^2 <--- reflects across x-axis, vertex (-3,0)
axis of symmetry x=-3
Subtracting 7 from right side
y = -4(x+3)^2 - 7 <--- shift 7 units downward, vertex (3,-7)
axis of symmetry x=-3
--------------------
Edwin
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