Power Functions

Power Functions

 

 

 

Power Function: Simplest form of a polynomial function in the form f(x) = ax^n

 

 

Key Features of a Graph

 

 

 

End Behavior: Behavior of the y-values as x increases to infinity and as x decreases to negative infinity.  This means, as the x values increase (i.e. 10, 100, 1000, 10000) what happens to the y-values ? If the corresponding y-values are (53, 420, 6404, 32000) then the y-values are increasing as the x –values are increasing. Therefore, “As x is approaching infinity, y is approaching infinity.” If the y-values were in a decreasing format, then the statement would be “As x is approaching infinity, y is approaching negative infinity”


Symmetry: A function could either have line or point symmetry. Even functions have symmetry in the y-axis (x=0). Odd functions have point symmetry about the origin (0,0) – meaning that if the function is rotated 180 degrees about the origin it will match up.

 

 

Example 1:

 

 

 

y = x

 

Domain: (XER)

Range: (YER)

X-intercept = 0

Y-intercept = 0

Degree = 1

Symmetry: Point  - Because if the function is reflected over the y-axis it wouldn't have symmetry, only if it is rotated 180 degrees, then it will match

End Behavior: As x approaches positive infinity, y approaches positive infinity. As x approaches negative infinity, y approaches negative infinity.

Extends From: Quadrant 3 to Quadrant 1

 

 

Example 2:

 

 

 

 

y = -x

 

Domain: (XER)

Range: (YER)

X-intercept = 0

Y-intercept = 0

Degree = 1

Symmetry: Point  - Because if the function is reflected over the y-axis it wouldn't have symmetry, only if it is rotated 180 degrees, then it will match

End Behavior: As x approaches positive infinity, y approaches negative infinity. As x approaches negative infinity, y approaches positive infinity.

Extends From: Quadrant 2 to Quadrant 4

 

Example 3:

 

 

 

 

y = ax^2 (where “a” can be any value, the properties will always be the same)

 

Domain: (XER)

Range: (YER | y >= 0)

X-intercept = 0

Y-intercept = 0

Degree = 2

Symmetry: Line  - Because if the function is reflected over the y-axis it would be symmetrical.

End Behavior: As x approaches positive infinity, y approaches positive infinity. As x approaches negative infinity, y approaches positive infinity.

Extends From: Quadrant 2 to Quadrant 1

 

Example 4:

 

 

y = -ax^2 (where “a” can be any value, the properties will always be the same)

 

Domain: (XER)

Range: (YER | y <= 0)

X-intercept = 0

Y-intercept = 0

Degree = 2

Symmetry: Line  - Because if the function is reflected over the y-axis it would be symmetrical.

End Behavior: As x approaches positive infinity, y approaches negative infinity. As x approaches negative infinity, y approaches negative infinity.

Extends From: Quadrant 3 to Quadrant 4

 

Example 5:

 

 

 

y = ax^3 (where “a” can be any value, the properties will always be the same)

 

Domain: (XER)

Range: (YER)

X-intercept = 0

Y-intercept = 0

Degree = 3

Symmetry: Point 

End Behavior: As x approaches positive infinity, y approaches positive infinity. As x approaches negative infinity, y approaches negative infinity.

Extends From: Quadrant 3 to Quadrant 1

 

 

Example 6:

 

 

 

 

y = -ax^3 (where “a” can be any value, the properties will always be the same)

 

Domain: (XER)

Range: (YER)

X-intercept = 0

Y-intercept = 0

Degree = 3

Symmetry: Point 

End Behavior: As x approaches positive infinity, y approaches negative infinity. As x approaches negative infinity, y approaches positive infinity.

Extends From: Quadrant 2 to Quadrant 4

 

 

 

 

Practice Question:

 

Without graphing, determine the key features of the following functions: f(x) = 3x^7 and g(x) = 8x^9

 

 

Click Here to See Answer


The domain is x belongs to all real numbers (XER), the range is y belongs to all real numbers (YER). The x-intercept is 0 and the y-intercept is 0. The degree of the functions are odd, therefore they have point symmetry. As x approaches positive infinity, y approaches positive infinity. As x approaches negative infinity, y approaches negative infinity.

 

 

 

 

If you have any questions, leave them in the comments below.