transformations of quadratics and absolute value graphs
Post on 13-Nov-2021
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Recall Vertex Form of Quadratic Functions
๐ ๐ฅ = ๐ ๐ฅ โ โ 2 + ๐
This form can be very helpful for graphing other transformed quadratics without having to find the
x and y-intercepts.
Think about how โhโ is the x-coordinate of the vertex.
โข This tells us how much the graph shifts left or right (horizontal shift)
โข Notice the โ h in the formula. This always shifts it opposite.
Think about how โkโ is the y-coordinate of the vertex.
โข This tells us how much the graph moves up or down (vertical shift)
Now we will learn about how the graph of quadratic functions can be stretched or compressed.
Transformations of Quadratic Functions
This form allows stretches/compressions as well as the vertical and horizontal shifts.
๐ ๐ฅ = ๐ ๐๐ฅ โ โ 2 + ๐
โaโ โ vertical stretch or compression and/or reflection
โข This will affect our y-coordinates (think vertical)
โข If a is negative, it will reflect across the x-axis
โbโ โ horizontal stretch or compression and/or reflection
โข This will affect our x-coordinates by the reciprocal (think horizontal is opposite)
โข If b is negative, it will reflect across the y-axis
Order of Transformations
When transforming graphs, you must transform in the following order:
1. Horizontal shifts (left and right)2. Stretches/compressions and Reflections3. Vertical shifts (up and down)
Example:
Tell what changes are made (in the correct order) to the graph of ๐ ๐ฅ = ๐ฅ2 to obtain each
graph:
1. ๐ ๐ฅ = โ ๐ฅ + 5 2 + 7 2. ๐ ๐ฅ = 3 ๐ฅ 2 โ 8 3. ๐ฆ =1
2๐ฅ + 1
2
Example:
Let ๐ ๐ฅ = ๐ฅ2, write a new function that translates ๐ ๐ฅ as described.
1. Vertical shrink of 1
3, left 5 units and up 2 units. 2. Horizontal stretch by 2, down 4 units
Transformations of Quadratic Functions
When transforming (translating) quadratic graphs, it is easiest to use the following special points:
0,0 1,1 (โ1,1)(2,4)(โ2,4)
Where do these points come from?
When we transform the quadratic equations, we will use these points and make the changes to
each x and y-coordinates.
Transformations of Quadratic Functions
๐ ๐ฅ = ๐ ๐๐ฅ โ โ 2 + ๐
This is how we will change our points with the transformations:
โhโ โ horizontal shift
โข Add or subtract to x-coordinates (opposite)
โaโ โ vertical stretch or compression
โข Multiply to the y-coordinates
โbโ โ horizontal stretch or compression
โข Multiply to the x-coordinates by the reciprocal (think horizontal is opposite)
โkโ โ vertical shift
โข Add or subtract to y-coordinates
Example: Change to vertex form, then draw the graph and label important points.
๐ ๐ฅ = โ๐ฅ2 โ 2๐ฅ โ 4
Transformations of Absolute Value Functions
๐ ๐ฅ = ๐ ๐๐ฅ โ โ + ๐
This is just like the quadratic functions equation, except it contains and absolute value instead of a
square. The a, b, h, and k still transform the absolute value graphs in the same way.
However, since it is a different function, it has different points: 0,0 1,1 (โ1,1)Where the points come from:
The shape is also different. Example of the shape:
Example:
Tell what changes are made (in the correct order) to the graph of ๐ ๐ฅ = ๐ฅ to obtain each
graph:
1. ๐ ๐ฅ = โ2 ๐ฅ + 1 โ 2 2. ๐ ๐ฅ = โ3๐ฅ โ 4
Example:
Let ๐ ๐ฅ = ๐ฅ , write a new function that translates ๐ ๐ฅ as described.
1. Horizontal shrink of 3, left 2 units. 2. Vertical shrink of 1/2 , reflect y-axis, and up 1 unit
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