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This page shows questions in the Algebra I Reasoning public release module at MSDE. Algebra 1
"Algebra I Reasoning"

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Expand All Official Practice Test

This is a multiple choice question that allows you to select only one option.

Consider the equation (x−a)2−b=0, where a and b are both positive real numbers.

Which statement about the solution or solutions to the equation must be true?

  1. A.
    The equation has no real solutions.
  2. B.
    The equation has exactly one real solution and the solution is positive.
  3. C.
    The equation has exactly two real solutions and both of the solutions are positive.
  4. D.
    The equation has exactly two real solutions and at least one of the solutions is positive.
i4

Correcti1

Thinks a square root of a negative number is takeni2

Only considers the positive square root of bi3

Thinks a must be greater than the square root of b

This is a test question that allows you to enter extended text in your response.

The system of equations shown is graphed in the xy-plane.

{x+y=12x-3y=17

Determine the coordinates of the solutions of the system algebraically. Then show that your solution is valid. Show your work or explain how you found your answer.

Enter your answer and your work or explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

There is no alt-text. (39 files were searched.)

This is a test question that allows you to enter extended text in your response.

The quadratic function f is defined as f(x)=-(x+2)2+3.

The quadratic functions g and h are transformations of the function f.

Function g is defined as g of x equals f of x plus k, where k is a constant.

Function h is defined as h of x equals f of open parenthesis x plus n close parenthesis, where n is a constant.

  • Determine whether there are values of k for which the graph of y equals g of x has no x-intercepts. Justify your answer.
  • Determine whether there are values of n for which the graph of y equals h of x has no x-intercepts. Justify your answer.

Enter your answers and your justifications in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

There is no alt-text. (39 files were searched.)

Holistic Rubric for 4-Point Reasoning Constructed Response Items

Sample Top Score Response

Part A

The graph of the function f is a parabola opening down with a vertex 3 units above the x –axis. Shifting the function down by more than 3 units would result in a graph with no x –intercepts. The transformation would be of the form g ( x ) = f ( x ) + k where k < − 3 .

Part B

There is no such transformation. The graph of f is a parabola with two x –intercepts and a domain of all real numbers. No matter how much the parabola is shifted to the left or right, there will always be two x –intercepts.

Points

Sample Characteristics

4 Points

A four-point response for reasoning items provides evidence of correct, complete, and appropriate mathematical reasoning.

The response may:

  • be clear and well developed with logical reasoning communicated by the use of precise and appropriate representations, symbols, drawings, or mathematical vocabulary.
  • demonstrate a thorough understanding of the mathematics.
  • contain minor flaws that do not detract from the correct reasoning or demonstration of a thorough understanding.

3 Points

A three-point response for reasoning items provides evidence of essentially correct, essentially complete, and essentially appropriate mathematical reasoning.

The response may:

  • be clear and developed with logical reasoning communicated by the use of essentially precise and appropriate representations, symbols, drawings, or mathematical vocabulary.
  • contain minor flaws that detract from the correct reasoning or demonstration of a thorough understanding.

2 Points

A two-point response for reasoning items provides evidence of partially correct mathematical reasoning.

The response may:

  • display an incomplete reasoning process.
  • contain mathematical flaws.

1 Point

A one-point response for reasoning items provides limited evidence of correct mathematical reasoning.

The response may:

  • demonstrate the beginning of a valid chain of reasoning.
  • reflect a lack of essential understanding of the underlying mathematical concepts.
  • contain the correct solution, but work is limited or missing.
  • contain errors in the fundamental mathematical procedures or reasoning.
  • contain omissions or irregularities that lead to an inadequate solution.

0 Point

A zero-point response is completely incorrect, incoherent or irrelevant.

This is a test question that allows you to select a spot or region on a graphic.

A system of inequalities is given, where a and b are positive integers.

{y<ax-by>-ax+b

Which region in the xy-plane shown contains solutions of the system?

Select one region.

This is a multiple choice question that allows you to select only one option.

A student claims that if an equation has no exponents greater than 1, then the equation is linear.

Which equation can be used to show that the student’s claim is false ?

  1. A.
    y=x
  2. B.
    y=|x|
  3. C.
    x+y=1
  4. D.
    y−x2=1
i2

Correcti1

Chooses an equation for which the student’s claim is true

Chooses an equation that is not proportional but that does not show that the student’s claim is false

Chooses an equation that does not show that the student’s claim is false because the equation contains an exponent greater than 1

This is a matching question that allows you to match elements from one list with those on another list.

The quadratic function f is graphed in the xy-plane shown.

The figure shows the graph of a function in the x y plane. The numbers negative 5 through 5, in increments of 1, are indicated on both axes. The function is in the shape of a parabola that opens upward. It has a vertex at the point with coordinates 1 comma negative 4. The parabola crosses the x axis at negative 1 and 3.

The function will be transformed to form a second function, g.

For each transformation of f, indicate whether the equation f(x)=g(x) has no solutions, one solution, or two solutions.

Select the boxes to show the number of solutions to each equation. Select only one box per row.

Transformation f(x)=g(x) Has No Solutions f(x)=g(x) Has One Solution f(x)=g(x) Has Two Solutions
g(x)=f(-x)
g(x)=-f(x)
g(x)=f(x)+k, where k is not equal to 0