The Journey into 6 different functions Family of Functions Linear Functions Quadratic Functions Cubic Functions Absolute Value Functions Radical Functions Rational Functions Parent Function Linear Notation-> f(x)=mx+b

- “m” is equal to the can stretch or shrink the graph by being equal to numbers other than 1.

- X is where the value of X is put.

- “b” is changed to make Vertical and Horizontal translations. Description: A linear function is a straight line that does not have multiple intervals in the line. A linear function could not have an up interval and a down interval. The slope of the line can be determined by the value of “m” and the y and x- intercepts can be determined by the value of “b”. f(x) = -x F(x) = x Reflection Reflection: The Y values are the opposite of the linear parent’s values. In the parent function, when X = 2, Y = 2. In the reflection of the parent function, when X = 2, Y = -2. The function can be reflected making “m” negative. The interval values are switched. This function is f(x) = -x. Shrink Shrink: If the line is steeper than the parent function than the absolute value of “m” is greater than 1. In this child graph, the value of “m” is 2. f(x)= 2x Stretched Stretch: If the line is shallower than the parent function than the absolute value of “m” is less than 1. In this child graph, the value of “m” is 1/2. f(x)= (1/2)x Vertical Translation Vertical Translation: The point at which the line crosses the y-axis is increased. The point can be raised by making “b” a positive number (see graph). It can also be decreased by making “b” a negative number. The function of this line is f(x) = x + 2. f(x)= x+2 Table: Linear (Parent)

f(x) = x

X Y

-5 -5

-4 -4

-3 -3

-2 -2

-1 -1

0 0

1 1

2 2

3 3

4 4

5 5 Domain = All Real Numbers

Range = All Real Numbers

x-intercept = 0

y-intercept = 0

Maximum = n/a

Minimum = n/a

Interval Up = -infinity∞ to infinity∞

Interval Down = n/a

End Behavior = The line will continue at a constant rate with a continuing straight line. Characteristics Horizontal Translation Horizontal Transition: The point at which the line crosses the x-axis is either increased or decreased. If the point where the line crosses the x-axis is increased, then “b” is a negative number. If the point where the line crosses the x-axis is decreased, then “b” is a positive number. The function of this line is f(x) = x 2. f(x)= x-2 Parent Function Quadratic Notation > f(x)=ax^2+bx+c or f(x)=(xd)^2 Description: The quadratic parabola resembles a “U” and will have Y values that are the same as another Y value in the line. The characteristics of the graph can be determined by the values of “a”, “b”, and “c”. The parabola is symmetrical. Table: Quadratic (Parent)

f(x) = x^2

X Y

-5 25

-4 16

-3 9

-2 4

-1 1

0 0

1 1

2 4

3 9

4 16

5 25

Characteristics

Domain = All Real Numbers

Range = 0

x-intercept = 0

y-intercept = 0

Maximum = n/a

Minimum = 0

Interval Up = 0 to infinity

Interval Down = -infinity to 0

End Behavior = The Y values will continue to increase Reflection Reflection: The Y values are the opposite of the quadratic parent’s values. In the parent function, when X = 2, Y = 4. In the reflection of the parent function, when X = 2, Y = -4. The parent function can be reflected if a negative sign is put in front of “a”. The line is mirrored from its peak. There is no minimum for this function and the maximum becomes 0. f(x)= -x^2 Shrink Shrink: The line can be shrunk by increasing the value of “a” to a number that is greater than 1 which means |“a”| > 1. The distance between two equal Y values is less in a shrunken quadratic line. “a” in this function is 2. f(x)= 2x^2 Stretched Stretch: The line can be stretched by decreasing the value of “a” to a number that is between 1 and 0 which means 0 < |“a”| < 1. The closer “a” is to 0, the more stretched it is going to be. The distance between two equal Y values is more in a stretched quadratic line. “a” in this function is ½. f(x)=(1/2)x^2 Vertical Translation Translated Vertically: The line can be translated vertically by making the value of “c” an integer that is not 0. In this function, the value of “c” is -4, and this can be figured out by looking graph and looking at the Y value when X = 0. The minimum is the value of “c” if the parabola is positive. If the parabola is negative then the value of “c” is the maximum. f(x)= x^2 + 4 Horizontal Translation Translated Horizontally: The line can be translated horizontally by using the function f(x)=a(x+d)^2. The value of “d” will move the parabola in reverse. In this function “d” = 1 so it moves left 1 unit. f(x)= (x+1)^2 Cubic Parent Cubic Notation -> f(x)=ax^3+bx^2+cx+d or f(x)= a(x + h)+k Description: The cubic function shows the value of X to the 3rd power. The simplest form of this function is x^3. The steepness of the slope of this line can be determined by the value of “a”. In the first notation listed, the value of d can determine where the y-intercept is. In the second notation, “k” determines where the y-intercept is if there is no “h” value. f(x)= x^3 Cubic (Parent)

f(x)=x^3

X Y

-5 -125

-4 -64

-3 -27

-2 -8

-1 -1

0 0

1 1

2 8

3 27

4 64

5 125

Table: Domain = All Real Numbers

Range = All Real Numbers

x-intercept = 0

y-intercept = 0

Maximum = n/a

Minimum = n/a

Interval Up = -infinity to infinity

Interval Down = n/a

End Behavior = The left side will continue going negative and the right side will continue being positive. Characteristics: Reflection Reflection: The cubic function reflected is the parent function flipped on the x-axis. The positive values are the same except that they are negative, and the negative values are the same except they are positive. The interval values are the opposite of the parent’s. f(x)= -x^3 Shrink Shrink: The cubic line can be shrunk by making the absolute value of “a” greater than 1. In this function the value of “a” is 2. All of the x^3 values are doubled in this function. f(x)= 2x^3 Stretched Stretched: The cubic line can be stretched by making the absolute value of “a” between 0 and 1. This means 0|"a" |1. The value of “a” in this function is . All of the x^3 values have been divided by two. f(x)= (1/2)x^3 Vertical Translation Vertical Translation: The cubic function can be translated vertically by making the value of “d” or “k”, depending on what notation is used, the number of units that is wanted to move. The value of “d” and “k” is -4. f(x)= X^3 - 4 Horizontal Translation Horizontal Translation: The cubic line can be translated horizontally by making the absolute value of “h” (in the second notation) a number greater than 0. The value of “h” will move the line in the reverse direction which means that negative values move it to the right and positive values move it to the left. The value of “h” in this function is 3. f(x)= (x + 3)^3 Parent Function Absolute Value Notation > f(x)=m|x+b|+c

- “m” is the slope of the lines.

- “b” is used to translate the function left and right but it is reversed.

-“c” is used to translate the function up and down.

-X is where the value of X is put into the function. Description: The absolute value line is a “V” and it gives the absolute of value of X and then can be changed with “m”, “b”, and “c”. The parent function makes a 90° angle on the graph. f(x)= |x| Absolute Value (Parent)

f(x) = |x|

X Y

-5 5

-4 4

-3 3

-2 2

-1 1

0 0

1 1

2 2

3 3

4 4

5 5

Domain = All Real Numbers

Range = All Real Numbers

x-intercept = 0

y-intercept = 0

Maximum = n/a

Minimum = 0

Interval Up = 0 to infinity

Interval Down = -infinity to 0

End Behavior = The line will continue at a constant rate with a continuing straight line in the positive direction.

Reflection Reflection: The line can be a reflection of the parent line by making “m” negative. The y-values for this function are the opposite of the y values that the parent has when the same X value is used. In this function, the maximum is 0 and there is no minimum. The y-values are all negative except for 0. f(x)= -|x| Shrink f(x)= 2|x| Shrink: The absolute value line can be shrunk by making “m” a value that has an absolute value greater than 1. |“m”| > 1. In this function, the value of “m” is 2. The distance between two equal y-values is less than in the parent function. Stretch f(x)= (1/2)|x| Stretch: The absolute value line can be stretched by making “m” a value that has an absolute value greater than 0 but less than 1. 0 < |“m”| < 1. The value of “m” in this function is ½. The distance between two equal y-values is greater than in the parent function. Vertical Translation Translated Vertically: An absolute value line can be translated vertically by changing the value of “c”. The value of “c” where the line is the number you will only see once. In this line, the value of “c” is -2. Also, if the value of “b” is 0, then the value of “c” is the y-intercept. f(x)= |x|-2 Horizontal Translation Translated Horizontally: An absolute value line can be translated horizontally by changing the value of “b”. The value of “b” moves the function inversely. If “b” = 2, the function would move 2 units to the left. If the value of “b” is a number other than 0, then the opposite of that value is the x-intercept if the value of “c” is 0. f(x)= |x+1| Parent Function Radical

f(x)= x^(1/2)

X Y

-5 n/a

-4 n/a

-3 n/a

-2 n/a

-1 n/a

0 0

1 1

2 1.4

3 1.7

4 2

5 2.2 Domain = x is greater than or equal to 0.

Range = y is greater than or equal to 0.

x-intercept = 0

y-intercept = 0

Maximum = n/a

Minimum = 0

Interval Up = 0 to infinity

Interval Down = n/a

End Behavior = The line will continue to be positive and will never have a Y value less than 0. Table: Characteristics: Radical Notation -> m(x+h)^(1/2) + k Description: The radical line represent 1/2 half of a parabola as if its line of symmetry were on the x-axis (with the origin being the vertex) and the negative side was cut off. The values for x and y of the Quadratic parent are switched with the radical values, which means that x is getting the perfect squares more often than y which is the opposite of the quadratic function. f(x)= x^(1/2) Reflection The reflection over the x-axis of the radical line is the negative half that the imaginary parabola makes with the x-axis being the line of symmetry and the origin being the vertex. f(x)= -x^(1/2) Shrink f(x)= 2(x)^(1/2) Stretched When stretching the line that the radical function makes, the value of "m" is changed to a number that has an absolute value greater than 1. The value of "m", in this case 2, will multiply to the square root of x to double the square root of x. When shrinking the line that the radical function makes, the value of"m" is changed to a value that has an absolute value between 0 and 1. Vertical Translation f(x)= (1/2)x^(1/2) f(x)= x^(1/2)+2 To translate the radical line vertically, the value of "k" must be the value at what height is wanted to be raised. If the value of "h" is zero, then the value of "k" is the y-intercept. The value of "k" is 2 in this function. Horizontal Translation To translate the line horizontally, the value of "h" must be greater than 0. The value of "h" and direction the line is moved is reacted inversely. In this function the value of "h" is 2 which means that the line is moved to the left 2 units. If the value of "k" is 0 then the value of "h" is the x-intercept. f(x)= (x+2)^(1/2) Parent Function Rational Notation -> f(x)= N(x)/D(x) Table: Rational

f(x)= 1/x

X Y

-5 -.2

-4 -.25

-3 -.33

-2 -.5

-1 -1

0 n/a

1 1

2 .5

3 .33

4 .25

5 .2 Description: This parent function cannot have a Y value when x is 0 because x is in the denominator in the function and 0 is undefined so there is no idea of what the answer is. The line skips 0. The X values that have the same absolute value give the same absolute value of y. When x = -2, y = -.5. When x = 2, y = .5. f(x)= 1/x Domain = All Real Numbers except for 0

Range = All Real Numbers except for 0

x-intercept = n/a

y-intercept = n/a

Maximum = n/a

Minimum = n/a

Interval Up = n/a

Interval Down = -infinity to infinity

End Behavior = The values of y will continue to get closer and closer to 0 but will never reach it. Characteristics:

Reflection When reflecting the parent function, a negative sign must be put to make the function negative. f(x)= -1/x Shrink When stretching a rational line, a number with an absolute value greater than 1 must be in the numerator. f(x)= 4/x Stretched When stretching a rational line, a number with an absolute value greater than 0 and less than 1 must be in the numerator. Vertical Translation When translating a rational line vertically, a number must be added to quotient after the division. This value added or subtracted from quotient will be the new horizontal asymptote. The horizontal asymptote in this line is 4. f(x)= (1/x)+4 Horizontal Translation When translating horizontally, a value must be added to the x to in a way that it can also divide the numerator. In this case, the number that I added to x was -2. The new vertical asymptote is 2 which means the value given affects the direction inversely. f(x)= 1/(x-2)

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# Function Family Album

Math 2: Phase 5 Project

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