Which Equation Represents A Linear Function Iready

Which Equation Represents a Linear Function i-Ready? A Comprehensive Guide

Understanding linear functions is crucial for success in algebra and beyond. This comprehensive guide will delve into identifying linear functions, exploring their equations, and providing a clear understanding of how to distinguish them from other function types, particularly within the context of i-Ready assessments. We'll tackle this by examining different equation forms, providing examples, and offering tips for solving related problems.

What is a Linear Function?

A linear function is a function that represents a straight line when graphed on a coordinate plane. This means that the rate of change (slope) between any two points on the line remains constant. The relationship between the input (x) and the output (y) is consistent and predictable. Linear functions are characterized by their simplicity and wide applicability in various fields, from physics to economics.

Key Characteristics of Linear Functions:

• Constant Rate of Change: The slope of the line is constant, meaning the y-value changes at a consistent rate for every unit change in the x-value.
• Straight Line Graph: When plotted on a coordinate plane, a linear function always produces a straight line.
• First-Degree Polynomial: The equation of a linear function is always a first-degree polynomial, meaning the highest exponent of the variable is 1.

Identifying Linear Functions from Equations

Several equation forms can represent a linear function. Understanding these forms is key to quickly identifying whether a given equation represents a linear function.

1. Slope-Intercept Form: y = mx + b

This is perhaps the most common form of a linear equation.

• y: Represents the dependent variable (the output).
• x: Represents the independent variable (the input).
• m: Represents the slope of the line (the rate of change). It indicates how steeply the line rises or falls. A positive 'm' indicates a positive slope (line goes up from left to right), while a negative 'm' indicates a negative slope (line goes down from left to right).
• b: Represents the y-intercept, the point where the line crosses the y-axis (when x = 0).

Example: y = 2x + 3 This equation represents a linear function with a slope of 2 and a y-intercept of 3.

2. Standard Form: Ax + By = C

This form is also commonly used to represent linear equations.

• A, B, and C: Are constants, with A and B not both zero.

Example: 3x + 2y = 6 This equation represents a linear function. To convert it to slope-intercept form, solve for y: 2y = -3x + 6, then y = (-3/2)x + 3.

3. Point-Slope Form: y - y1 = m(x - x1)

This form is useful when you know the slope (m) and a point (x1, y1) on the line.

• m: Represents the slope.
• (x1, y1): Represents a point on the line.

Example: y - 2 = 4(x - 1) This equation represents a linear function. It can be rewritten in slope-intercept form: y = 4x - 2.

Identifying Non-Linear Functions

It's equally important to recognize equations that do not represent linear functions. These functions often involve exponents other than 1, roots, or absolute values.

Examples of Non-Linear Functions:

• Quadratic Functions: These functions have an x² term (e.g., y = x² + 2x + 1). Their graphs are parabolas, not straight lines.
• Exponential Functions: These functions have the variable in the exponent (e.g., y = 2ˣ). Their graphs are curves, not straight lines.
• Radical Functions: These functions involve square roots or other roots (e.g., y = √x). Their graphs are curves, not straight lines.
• Absolute Value Functions: These functions involve the absolute value (e.g., y = |x|). Their graphs are V-shaped, not straight lines.

Solving i-Ready Style Problems

i-Ready assessments often present multiple-choice questions requiring you to identify which equation represents a linear function. Here's a step-by-step approach:

1. Look for the Highest Exponent: If the highest exponent of the variable (x) is 1, it's likely a linear function. If the exponent is higher than 1 (e.g., x², x³), it's not a linear function.

2. Check for Non-Linear Elements: Examine the equation for any square roots, absolute values, or variables in the exponent. The presence of these elements indicates a non-linear function.

3. Convert to Slope-Intercept Form (if necessary): If the equation isn't already in slope-intercept form (y = mx + b), try to rearrange it to this form. If you can successfully rearrange it into this form, it's a linear function.

4. Graph the Equation (if possible): If you're unsure, graphing the equation can visually confirm whether it produces a straight line.

5. Eliminate Incorrect Options: Use the above steps to eliminate options that clearly represent non-linear functions. This will help you narrow down your choices.

Example i-Ready Style Question:

Which of the following equations represents a linear function?

a) y = x² + 5 b) y = 3x - 2 c) y = √x d) y = 2ˣ

Solution:

• a) y = x² + 5: This is a quadratic function (because of x²) and therefore not linear.
• b) y = 3x - 2: This is in slope-intercept form, with a slope of 3 and a y-intercept of -2. This is a linear function.
• c) y = √x: This is a radical function and therefore not linear.
• d) y = 2ˣ: This is an exponential function and therefore not linear.

Therefore, the correct answer is b).

Advanced Considerations: Vertical and Horizontal Lines

While generally following the rules above, remember these special cases:

• Vertical Lines: A vertical line has the equation x = c, where 'c' is a constant. Although it's a straight line, it's not a function because it fails the vertical line test (multiple y-values for a single x-value).

• Horizontal Lines: A horizontal line has the equation y = c, where 'c' is a constant. This is a function because it passes the vertical line test. The slope is 0.

Understanding these exceptions is important for comprehensive understanding of linear functions.

Conclusion: Mastering Linear Functions in i-Ready

Identifying equations that represent linear functions is a fundamental skill tested frequently in i-Ready assessments and throughout algebra. By understanding the characteristics of linear functions, their various equation forms, and how to differentiate them from non-linear functions, you can confidently tackle these questions and achieve success. Remember to practice regularly and apply the steps outlined above to build your skills and master this essential mathematical concept. Consistent review and practice will solidify your understanding, ensuring you're well-prepared for any challenges i-Ready throws your way.