Domain II: Patterns and Algebra
The Patterns and Algebra domain has about 13 questions, accounting for 11% of the And that is some information about the TExES Math & Science 4-8 test. There are four competencies within this domain. Concepts covered within these competencies include:
- Patterns
- Functions
- Advanced Functions
- Calculus Connections
Let’s explore a few important specific topics that are likely to appear on the test.
Solving Systems of Linear Inequalities
A linear inequality is similar to a linear equation, except it lacks an equal sign. It represents a line and contains both x and y values and a symbol of inequality – >, <, ≥, ≤. The correct answer is not a straight line, but a section of the graph above or below the line.
For example, solve this system of linear inequalities:
Solution: The filled-in area underneath the lines is the solution. Any coordinate point in this area will satisfy all 3 inequalities. Note: The lines stop at the y-axis because the last inequality shows x > 0.
Check the work, using a point in the shaded area such as (4, -2)
3x + 4y < 16
3x – 2y > 8
x > 0
First equation: 3(4) + 4(-2) < 16 simplifies to 12 + -8 < 16 or 4 < 16
Second equation: 3(4) – 2(-2) > 8 simplifies to 12 + 4 > 8 or 16 > 8
Third Equation: 4 > 0
Basic Characteristics of Quadratic Functions
A quadratic function is an equation written as ax² + bx + c = y . When graphed, it forms a parabola, or “U” shape. A parabola is a symmetrical shape that points upward if the value of a is positive and downward if a is negative. Important features of a parabola can be found using the equation, such as the roots, vertex, and the axis of symmetry.
The roots are the points where the parabola intersects the x-axis. To find the roots of a quadratic function, set y = 0. Then solve for x.
Another feature of a parabola is the vertex, which is the bottom point of an upward-facing parabola or the top point of a downward-facing parabola. The formula to find the vertex is to find point (-b/2a, a(-b / 2a)² + b(-b / 2a) + c).
Another option is to find x by using -b/2a, and then substituting the value of x into the original formula to solve for y.
The axis of symmetry is an imaginary line that divides the parabola into congruent halves. The formula for the axis of symmetry is x = -b / 2a because it is a vertical line.
Example: If given the equation 3x² + 2x – 8 = y, identify the roots, vertex, and axis of symmetry.
Identifying the roots:
Set y = 0: 3x² + 2x – 8 = 0
Factor the equation: (3x – 4)(x + 2) = 0
Set each part equal to 0: 3x – 4 = 0 and x + 2 = 0
Solve: x = 4/3 and x = -2
The roots are 4/3 and -2
Identifying the vertex:
Use the formula: (-b/2a, a(-b / 2a)² + b(-b / 2a) + c)
Find the x value by substituting the formula -b/2a: -2/2(3)
Simplify: x = -⅓
Find the y value by substituting for x: 3(-⅓)² + 2(⅓) – 8 = y
Simplify: 3(⅑) + ⅔ – 8 = y
Simplify: ⅓ + ⅔ – 8 = y
Simplify: 1 – 8 = y
The vertex is (-⅓, -7)
Identifying the axis of symmetry:
Using the information from the vertex: -b/2a = -⅓
The axis of symmetry is x = -⅓
Rate of Change
There are two types of rate of change to know – average and instantaneous. The average rate of change is the slope of a curved line over a given interval. It is identified by a secant line, one that intersects the curve at least twice. Instantaneous rate of change is the change at a specific point and is identified by the tangent line, one that touches a curve at a point without crossing over it.
This is one of several concepts that can be introduced and related to calculus in middle school to build a foundation for advanced mathematics. Students learn how to identify the slope of a straight line and interact with curved lines. These concepts can be connected to calculus by teaching about the secant and tangent line. Students may not understand how to create these lines, but if they can identify them on a diagram and find their slope, this prepares them to connect the concepts to calculus later on.
And that is some basic information about Domain II of the TExES Math & Science 4-8 exam.