List a sequence of transformations which maps the graph of y=x^2 to the image with equation y=4*(2x-3)^2 + 1
The general transformation equation is:
g(x)=a*f(b(x-h))+k
where
h=horizontal translation (right if h>0)
b=horizontal scaling (stretch if 1/b>1)
a=vertical scaling (stretch if a>1)
k= vertical translation (up if k>0)
applied in the above order, i.e. according to the PEMDAS rule.
So the transformed equation can be written as:
let f(x)=x²
then
g(x)=4*(2x-3)² + 1
=4*(2(x-3/2))² +1
=4*f(2(x-3/2))+1
Identify the values of h,b,a,k as given above and hence the sequence of transformations.
Well, to go from y=x^2 to y=4*(2x-3)^2 + 1, we can follow these steps:
1. Start with y=x^2.
2. Multiply the x-values by 2 and shift the graph 3 units to the right: y=(2x-3)^2.
3. Multiply the y-values by 4: y=4*(2x-3)^2.
4. Finally, shift the graph 1 unit up: y=4*(2x-3)^2 + 1.
So, the sequence of transformations would be: stretch vertically by 4, shrink horizontally by a factor of 2, shift 3 units to the right, and shift 1 unit up.
To map the graph of y=x^2 to the image with equation y=4*(2x-3)^2 + 1, you need to perform the following transformations:
1. Horizontal Translation: The expression (2x-3) inside the parentheses indicates a horizontal translation. The graph is shifted horizontally to the right by 3 units.
2. Vertical Scaling: Multiplying the expression by 4 indicates a vertical scaling. The graph is stretched vertically by a factor of 4.
3. Vertical Translation: Adding 1 to the expression indicates a vertical translation. The graph is shifted vertically upward by 1 unit.
Therefore, the sequence of transformations is as follows:
1. Horizontal Translation: Right 3 units
2. Vertical Scaling: Stretch vertically by a factor of 4
3. Vertical Translation: Upward 1 unit
To map the graph of y = x^2 to the image with equation y = 4*(2x - 3)^2 + 1, we need to perform a sequence of transformations. Here are the steps to achieve this transformation:
Step 1: Horizontal stretch/compression
The expression 2x - 3 is inside the parentheses of (2x - 3)^2. This means a horizontal transformation is involved. Since the original equation has x^2, which represents a parabola with a vertical line of symmetry, we need to stretch or compress the parabola horizontally. The factor of 2 indicates a horizontal compression.
y = x^2 is transformed to y = (2x)^2.
Step 2: Horizontal shift
The expression (2x - 3) represents a horizontal shift. The constant -3 indicates a shift to the right by 3 units.
y = (2x)^2 is transformed to y = (2(x + 3/2))^2.
Step 3: Vertical stretch/compression
The coefficient 4 in front of the equation indicates a vertical stretch or compression. In this case, it is a vertical stretch.
y = (2(x + 3/2))^2 is transformed to y = 4(2(x + 3/2))^2.
Step 4: Vertical shift
Finally, the constant term 1 at the end of the equation represents a vertical shift. In this case, it is a vertical shift upward by 1 unit.
y = 4(2(x + 3/2))^2 is transformed to y = 4(2(x + 3/2))^2 + 1.
Therefore, the sequence of transformations that maps the graph of y = x^2 to the image with equation y = 4*(2x - 3)^2 + 1 is horizontal compression by a factor of 2, horizontal shift to the right by 3 units, vertical stretch by a factor of 4, and vertical shift upward by 1 unit.