This handout is a summary of Functions and Transformations
-Unit #3 from Mr. Quenneville’s Period 5 class.
General Terms
Relation
A relation is the relationship between two variables which can be expressed as ordered pairs (x, y), a table of values, through graphs or equations.
See The Vertical Line Test
Function
A function is a set of ordered pairs where for every value of x, there is only one value of y. Warning: All functions are relations, but not all relations are functions.
See The Vertical Line Test
Parent Function
A parent function is the beginning of any transformation. There are four parent functions from this unit: Quadratic, Radical, Reciprocal, and Absolute Value.
See examples on pages 4 +
Inverse Function
An inverse function has all the same points as the parent function, however the x and y coordinates are reversed.
Instead of it being f(x), it would be: f(x)−1 =
√
xAnd instead of a point being (3, 0), it would be: (0, 3)
Steps:
1. Replace f(x) with y 2. Switch the x and y’s 3. Solve for y (Isolation)
Domain
The domain of a function is the set of values of the independent variable. (Across the X Axis)
D: {XER}
D: {XER / x (<, ≤ ,≥, >) __ }
What is inside the brackets are symbols that represent the restrictions applied to the Domain:
If x is less than (<) a certain number, then there are no x values that go beyond that number.
If x is greater than (>) a certain number, then there are no x values that are below that number.
If x is less than or equal to (≤) a certain number, then x shares the same value as that number, though there are no x values that go beyond it.
If x is greater than or equal to (≥) a certain number, then x shares the same value as that number, though there are no x values that are below it.
Range
The range of a function is the set of values of the dependent variable. (Up and down the Y Axis)
R: {YER}
R: {YER / y (<, ≤ ,≥, >) __ }
What is inside the brackets are symbols that represent the restrictions applied to the Range:
If y is less than (<) a certain number, then there are no y values that go beyond that number.
If y is greater than (>) a certain number, then there are no y values that are below that number.
If y is less than or equal to (≤) a certain number, then y shares the same value as that number, though there are no y values that go beyond it.
If y is greater than or equal to (≥) a certain number, then y shares the same value as that number, though there are no y values that are below it.
Transformation
A transformation is a change made to the parent function that moves it from one position to another. The variables in an equation that cause transformations are known as: a, k, d and c.
**Everything inside the brackets transform the x coordinates, and everything outside transform the y coordinates.**
f
(
x)
=a[
k(
x−d)
]
+c The a value:When ‘a’ is + , the function is not reflected in the x axis. When ‘a’ is -, the function is reflected in the x axis.
If ‘a’ < 1, the function is vertically compressed by a factor of ‘a’. (A fraction value)
The k value:
When ‘k’ is +, the function is not reflected in the y axis. When ‘k’ is -, the function is reflected in the y axis.
If ‘k’ > 1, the function is horizontally stretched by a factor of ‘k’. (Not a fraction value) If ‘k’ < 1, the function is horizontally compressed by a factor of ‘k’. (A fraction value)
The d value:
When ‘d’ is – in the equation, the function is translated to the right. When ‘d’ is + in the equation, the function is translated to the left.
The c value:
When ‘c’ is – in the equation, the function is translated down. When ‘c’ is + in the equation, the function is translated up.
Also see The RST Chart
Invariant Point
An invariant point is a point that does not change after a transformation.
For example: f(x) = x and one of its points is (0, 0) & g(x) = 2x also has (0, 0) as one of its points *That would mean it is an invariant point!
The Vertical Line Test
The vertical line test is a method of determining whether a relation on a graph is a function or not. **If a vertical line passes through more than one point on the graph, the relation is not a function.
Ordered Pair
An ordered pair is a pair of numbers that are used to locate a point on the graph. (coordinates)
(x, y)
(1, -4)
*This is just an example
The RST Chart
The RST chart is a chart with six boxes that contain information that is applicable to graphing a function’s transformations. It makes everyone’s graphing lives easier.
R = Reflections
T = Translations
The Fantastic Four
The Quadratic Function – Also known as the parabolic function
f (x) = x2
*KEY POINTS
(-2, 4) (-1, 1) (0, 0) (1, 1) (2, 4)
Domain and Range
D: {XER}
R: {YER / y ≥ 0}
The Radical Function – Also known as the square root function
f (x) =
√
x*KEY POINTS
(0, 0) (1, 1) (4, 2) (9, 3)
Domain and Range
D: {XER / x ≥ 0}
R: {YER / y ≥ 0}
The Reciprocal Function – Also known as the fraction function
f (x) = 1
x
*KEY POINTS
(1
2, 2)
(1, 1)
(2, 1
2)
(−1
(-1, -1)
(-2, −1
2 )
Asymptote
An Asymptote is a line that of which the functions cannot ever cross.
There are known two types of Asymptotes from this unit: Horizontal, and Vertical
Domain and Range
D: {XER }
R: {YER / y ≥ 0}
The Absolute Value Function
f (x) = | x |
*KEY POINTS (2, 2)
(1, 1) (0, 0) (-1, 1) (-2, 2)
Domain and Range
D: {XER }
R: {YER / y ≥ 0}
Tips for Drawing Graphs:
Values on the X and Y axis are labelled and are easy to read ✓
✓The graph has a title
✓The function drawn is labelled with its equation as well as at least three points, including the vertex
✓Remember to draw arrows at the end of lines that would go on continuously, such as the ends of the X and Y axis or the end of a radical function
✓If the function is reciprocal, remember to label the asymptotes
