Math
Spring Operational 2015
Algebra 2
PBA Item #18
Property of an Even Function
2092-M40742
Prompt
Rubric
Task is worth a total of 4 points.
2092-M40742 Rubric Part A
Score Description
1
Student response includes the following element.
 Reasoning component = 1 point
o Valid explanation of a characteristic of an even function
shown in the graph
Sample Student Response:
If P is an even function, then P ( x ) would equal P (x ) for all
values of x in the domain of P. Graphically, this means that the
graph is symmetric about the y-axis. The graph given seems to
show this.
0
Student response is incorrect or irrelevant.
2092-M40742 Rubric Part B
Score Description
3
Student response includes the following 3 elements.
 Reasoning component = 3 points
o Correct algebraic definition of an even function
o Correct application of definition
o Correct work showing that the function is even
Sample Student Response:
If P (x)  P(x), then P is an even function.
4
2
P ( x )  ( x )  8( x )  16
 x 4  8x 2  16
 P (x)
Scoring note: The second scoring element, applying the definition
correctly, will likely be shown in the first line of the student’s work,
where the student writes P ( x )  ( x ) 4  8( x ) 2  16.
2
Student response includes 2 of the 3 elements.
1
Student response includes 1 of the 3 elements.
0
Student response is incorrect or irrelevant.
Anchor Set
A1 – A10
A1
Part A: Score Point 1
Part B: Score Point 3
Annotations
Anchor Paper 1
Part A: Score Point 1
This response receives full credit. The student includes the required element.

The student includes an explanation of a characteristic of an even function shown in
the graph (Symmetry across the y-axis).
Part B: Score Point 3
This response receives full credit. The student includes each of the three required elements.

The student includes a correct algebraic definition of an even function (p(x) =p(-x)).

The student includes an application of the definition (p(-x) = (-x)4 – 8(-x)2 +16).

The student includes correct work showing that the function is even
(p(-x) = (-x)4 – 8(-x)2 +16; x4 – 8x2 +16 = x4 – 8x2 +16).
A2
Part A: Score Point 1
Part B: Score Point 3
Annotations
Anchor Paper 2
Part A: Score Point 1
This response receives full credit. The student includes the required element.

The student includes an explanation of a characteristic of an even function shown in
the graph (the reflection across the y axis). [Note: Also acceptable is, (for any x
coordinate, the y is the same for –x).]
Part B: Score Point 3
This response receives full credit. The student includes each of the three required elements.

The student includes a correct algebraic definition of an even function (f(x) = f(-x)).

The student includes an application of the definition ((-x)4 – 8(-x)2 +16).

The student includes correct work showing that the function is even
(x4 – 8x2 +16 = (-x)4 – 8(-x)2 +16; x4 – 8x2 +16 = x4 – 8x2 +16).
A3
Part A: Score Point 0
Part B: Score Point 3
Annotations
Anchor Paper 3
Part A: Score Point 0
This response receives no credit. The student does not include the required element.
The response includes an insufficient explanation of a characteristic of an even function
shown in the graph (The graph infinitely increases on both the right and left side. This
behavior is only seen in even functions). This statement is only true for positive even
functions, not all even functions, when describing the end behaviors of the function.
Part B: Score Point 3
This response receives full credit. The student includes each of the three required elements.

The student includes a correct algebraic definition of an even function (P(-x) = x4 –
8x2 + 16 = P(x)).

The student includes an application of the definition (P(-x) = (-x)4 – 8(-x)2 + 16).

The student includes correct work showing that the function is even
(P(-x) = (-x)4 – 8(-x)2 + 16; (-x)4 = x4; (-x)2 = x2; P(-x) = x4 – 8x2 + 16 = P(x)).
A4
Part A: Score Point 0
Part B: Score Point 3
Annotations
Anchor Paper 4
Part A: Score Point 0
This response receives no credit. The student does not include the required element.
The response includes an insufficient explanation of a characteristic of an even function
shown in the graph (the graph rises to the left and rises to the right as x approaches either
positive or negative infinity). This statement is only true for positive even functions, not all
even functions, when describing the end behaviors of the function.
Part B: Score Point 3
This response receives full credit. The student includes each of the three required elements.

The student includes a correct algebraic definition of an even function (P(x) = P(-x)).

The student includes an application of the definition (P(-x) = (-x)4 – 8(-x)2 + 16).

The student includes correct work showing that the function is even
(P(-x) = (-x)4 – 8(-x)2 + 16; P(-x) = x4 – 8x2 + 16; P(x) = x4 – 8x2 +16).
A5
Part A: Score Point 0
Part B: Score Point 2
Annotations
Anchor Paper 5
Part A: Score Point 0
This response receives no credit. The student does not include the required element.
The response includes an insufficient explanation of a characteristic of an even function
shown in the graph (the graph will move up as the x values decrease infinitely, and the graph
will move up as the x value increases infinitely). This statement is only true for positive even
functions, not all even functions, when describing the end behaviors of the function.
Part B: Score Point 2
This response receives partial credit. The student includes two of the three required
elements.

The student includes a correct algebraic definition of an even function (f(x) = f(-x)).

The student includes an application of the definition ((-x)4 – 8(-x)2 + 16).
The response includes correct work showing that the function is even with an incorrect
statement (x4 – 8x2 + 16 = (-x)4 – 8(-x)2 + 16; The powers of x throughout the equation are
positive, therefore making this equation the same whether x is positive or negative. So the
equation must be positive). The statement is not true because P(x) = x-4 – 8x-2 +16 is an
even function as well. Nor do the exponents have to be even because P(x) = (cos)x with an
exponent of 1 is an even function as well.
A6
Part A: Score Point 1
Part B: Score Point 1
Annotations
Anchor Paper 6
Part A: Score Point 1
This response receives full credit. The student includes the required element.

The student includes an explanation of a characteristic of an even function shown in
the graph (symmetry across the y axis).
Part B: Score Point 1
This response receives partial credit. The student includes one of the three required
elements.

The student includes a correct algebraic definition of an even function (a function is
even if you plug –x in for x, and you get the same equation after you simplify).
The response does not include an application of the definition.
The response includes incorrect work showing that the function is even
(-x4 – 8(-x)2 + 16; x2 – 8x + 16). The exponents in the second expression are not correct.
A7
Part A: Score Point 1
Part B: Score Point 0
Annotations
Anchor Paper 7
Part A: Score Point 1
This response receives full credit. The student includes the required element.

The student includes an explanation of a characteristic of an even function shown in
the graph (symetrical over the y axis).
Part B: Score Point 0
This response receives no credit. The student includes none of the three required elements.
The response includes an incorrect algebraic definition of an even function (Where the
graphed line is symetrical over a line sof symetry or the axis). This is a graphical definition of
an even function, not an algebraic definition.
The response does not include an application of the definition.
The response does not include work showing that the function is even.
A8
Part A: Score Point 1
Part B: Score Point 0
Annotations
Anchor Paper 8
Part A: Score Point 1
This response receives full credit. The student includes the required element.

The student includes an explanation of a characteristic of an even function shown in
the graph (The function is a reflection over the y axis).
Part B: Score Point 0
This response receives no credit. The student includes none of the three required elements.
The response includes an incorrect algebraic definition of an even function (an even function
is a function that reflects over the x axis). An even function reflects over the y-axis, not the
x-axis, and this is an attempt at a graphical definition of an even function, not an algebraic
definition.
The response does not include an application of the definition.
The response does not include work showing that the function is even.
A9
Part A: Score Point 0
Part B: Score Point 0
Annotations
Anchor Paper 9
Part A: Score Point 0
This response receives no credit. The student does not include the required element.
The response does not include an explanation of a characteristic of an even function shown in
the graph (The output values are the same for positive and negative input values). This is
considered vague since it cannot be assumed that x is the input.
Part B: Score Point 0
This response receives no credit. The student includes none of the three required elements.
The response includes an incorrect algebraic definition of an even function (An even function
has the same output values for negative and positive input values). This is considered vague
as the response does not clarify that (negative and positive input values) are negative and
positive values of x.
The response does not include an application of the definition.
The response does not include work showing that the function is even.
A10
Part A: Score Point 0
Part B: Score Point 0
Annotations
Anchor Paper 10
Part A: Score Point 0
This response receives no credit. The student does not include the required element.
The response does not include an explanation of a characteristic of an even function shown in
the graph (It is symetrical). This is considered vague since the response does not mention
that the graph is symmetrical about the y-axis.
Part B: Score Point 0
This response receives no credit. The student includes none of the three required elements.
The response includes an incorrect algebraic definition of an even function (it has the
difference between two squares). This is not the algebraic definition of an even function.
The response does not include an application of the definition.
The response does not include work showing that the function is even.
Practice Set
P101 - P105
P101
P102
P103
P104
P105
Practice Set
Paper
Score
P101
0,0
P102
0,0
P103
1,3
P104
1,2
P105
1,1