transformations of quadratics and absolute value graphs
TRANSCRIPT
Recall Vertex Form of Quadratic Functions
π π₯ = π π₯ β β 2 + π
This form can be very helpful for graphing other transformed quadratics without having to find the
x and y-intercepts.
Think about how βhβ is the x-coordinate of the vertex.
β’ This tells us how much the graph shifts left or right (horizontal shift)
β’ Notice the β h in the formula. This always shifts it opposite.
Think about how βkβ is the y-coordinate of the vertex.
β’ 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.
π π₯ = π ππ₯ β β 2 + π
βaβ β vertical stretch or compression and/or reflection
β’ This will affect our y-coordinates (think vertical)
β’ If a is negative, it will reflect across the x-axis
βbβ β horizontal stretch or compression and/or reflection
β’ This will affect our x-coordinates by the reciprocal (think horizontal is opposite)
β’ 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 π π₯ = π₯2 to obtain each
graph:
1. π π₯ = β π₯ + 5 2 + 7 2. π π₯ = 3 π₯ 2 β 8 3. π¦ =1
2π₯ + 1
2
Example:
Let π π₯ = π₯2, write a new function that translates π π₯ 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 (β1,1)(2,4)(β2,4)
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
π π₯ = π ππ₯ β β 2 + π
This is how we will change our points with the transformations:
βhβ β horizontal shift
β’ Add or subtract to x-coordinates (opposite)
βaβ β vertical stretch or compression
β’ Multiply to the y-coordinates
βbβ β horizontal stretch or compression
β’ Multiply to the x-coordinates by the reciprocal (think horizontal is opposite)
βkβ β vertical shift
β’ Add or subtract to y-coordinates
Example: Change to vertex form, then draw the graph and label important points.
π π₯ = βπ₯2 β 2π₯ β 4
Transformations of Absolute Value Functions
π π₯ = π ππ₯ β β + π
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 (β1,1)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 π π₯ = π₯ to obtain each
graph:
1. π π₯ = β2 π₯ + 1 β 2 2. π π₯ = β3π₯ β 4
Example:
Let π π₯ = π₯ , write a new function that translates π π₯ as described.
1. Horizontal shrink of 3, left 2 units. 2. Vertical shrink of 1/2 , reflect y-axis, and up 1 unit