Question

Which equation does NOT implicitly define a linear function? A. \(y=x^2+5\) B. \(y=3 x+2\) C. \(4 x-6 y=11\) D. \(y+3=-2(x-5)\)

Short answer

Option A: \( y = x^2 + 5 \)

Step-by-step solution

Understand the Definition of a Linear Function

A linear function is one that can be written in the form: \( y = mx + b \), where \( m \) and \( b \) are constants.

Examine Option A

Option A: \( y = x^2 + 5 \). This is a quadratic equation because it includes \( x^2 \). Quadratic functions are not linear.

Examine Option B

Option B: \( y = 3x + 2 \). This is in the form \( y = mx + b \), where \( m = 3 \) and \( b = 2 \). This is a linear function.

Examine Option C

Option C: \( 4x - 6y = 11 \). This can be rewritten as \( y = \frac{2}{3}x - \frac{11}{6} \), which is in the form \( y = mx + b \). This is a linear function.

Examine Option D

Option D: \( y + 3 = -2(x - 5) \). This can be rewritten as \( y = -2x + 10 - 3 \), or \( y = -2x + 7 \). This is in the form \( y = mx + b \). This is a linear function.

Identify the Non-Linear Function

Options B, C, and D all represent linear functions, while Option A represents a quadratic function. Thus, Option A does not represent a linear function.

Key concepts

quadratic functions

Quadratic functions are a fundamental concept in algebra. They are characterized by the presence of an \( x^2 \) term. The general form of a quadratic function is given by: \[ y = ax^2 + bx + c \], where \( a \), \( b \), and \( c \) are constants. Unlike linear functions, which graph as straight lines, quadratic functions graph as parabolas. These parabolas can either open upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)).

Here are the key features of quadratic functions:

• **Vertex**: The highest or lowest point on the graph, given by the formula: \[ x = -\frac{b}{2a} \]
• **Axis of Symmetry**: A vertical line that passes through the vertex. It divides the parabola into two symmetrical halves.
• **Roots or Zeros**: The points where the parabola intersects the x-axis. These can be found using various methods like factoring, completing the square, or the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Quadratic functions play an essential role in many real-world applications, such as physics, engineering, and economics, where they can model phenomena like projectile motion and optimal pricing strategies.

function forms

Understanding different function forms is crucial for identifying and solving equations. One of the simplest forms is the linear function, typically written as: \[ y = mx + b \], where \( m \) is the slope and \( b \) is the y-intercept.

However, functions can take various forms, such as:

• **Quadratic Functions**: \[ y = ax^2 + bx + c \]
• **Cubic Functions**: \[ y = ax^3 + bx^2 + cx + d \]
• **Exponential Functions**: \[ y = a \cdot e^{bx} \]
• **Logarithmic Functions**: \[ y = a \cdot \log_b(x) + c \]

Each type of function has a unique graph and set of properties. For instance, while linear and quadratic functions have predictable and consistent forms, cubic functions can have more complex shapes with multiple turning points. Learning to recognize these forms helps in understanding their behavior and applications.

Special attention should be paid to the ability to manipulate these forms. For example, converting from one form to another, such as from a standard form to a factored form, can provide deeper insights into the function's properties, like its roots and intercepts.

GED math reasoning

The GED (General Educational Development) test includes a math reasoning section that assesses your understanding of various mathematical concepts. This part of the test covers topics like:

• **Algebra and Functions**: Includes linear equations, quadratic functions, and function forms
• **Geometry**: Measurement, shapes, and spatial reasoning
• **Number Operations**: Basic arithmetic, fractions, percentages, and ratios
• **Data Analysis**: Interpreting graphs, tables, and statistical measures

Mastery in GED math reasoning entails not just solving problems but understanding the underlying concepts. For instance, recognizing whether an equation represents a linear function or a quadratic function helps in solving it correctly.

Practice is key to success in math reasoning. Working on sample problems and taking practice tests can significantly boost your confidence and performance. Remember to take your time to understand each problem, break it down into smaller parts, and apply the appropriate mathematical tools and formulas. This approach will help you effectively tackle a wide range of questions on the GED maths reasoning section.