Identify Linear, Quadratic, and Exponential Functions QUIZ | 100 ... - Free Printable

The image you provided is a comparison table of three types of functions: Linear, Quadratic, and Exponential. Each column describes the characteristics of these functions in terms of their equations, graphs, and how to identify them from a table. Below, I will explain each section in detail:

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1. Linear Functions

#### Equation
- General form: \( y = mx + b \)
- \( m \): slope (rate of change)
- \( b \): y-intercept (where the line crosses the y-axis)
- Standard form: \( ax + by = c \)
- No exponent on \( x \) or \( y \).

#### Graph (Shape)
- The graph is a straight line.
- It can be increasing (positive slope), decreasing (negative slope), or horizontal (zero slope).

#### Identify from a Table
- In a table of values:
- The difference between consecutive \( y \)-values is constant.
- This constant difference indicates a linear relationship.

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2. Quadratic Functions

#### Equation
- General form: \( y = ax^2 + bx + c \)
- \( a \): coefficient of \( x^2 \)
- \( b \): coefficient of \( x \)
- \( c \): constant term
- Vertex form: \( y = a(x-h)^2 + k \)
- \( (h, k) \): vertex of the parabola
- Standard form: \( ax^2 + bx + c = 0 \)

#### Graph (Shape)
- The graph is a parabola (U-shaped curve).
- If \( a > 0 \), the parabola opens upwards (minimum point at the vertex).
- If \( a < 0 \), the parabola opens downwards (maximum point at the vertex).

#### Identify from a Table
- In a table of values:
- The second differences (differences of the differences) are constant.
- This constant second difference indicates a quadratic relationship.

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3. Exponential Functions

#### Equation
- General form: \( y = a(b)^x + c \)
- \( a \): initial factor (multiplier)
- \( b \): base (growth/decay factor)
- \( c \): vertical shift (asymptote)
- If \( b > 1 \), the function represents exponential growth.
- If \( 0 < b < 1 \), the function represents exponential decay.

#### Graph (Shape)
- The graph is a curved line.
- For exponential growth (\( b > 1 \)): The curve rises rapidly.
- For exponential decay (\( 0 < b < 1 \)): The curve approaches a horizontal asymptote as \( x \) increases.
- The horizontal asymptote is determined by the value of \( c \).

#### Identify from a Table
- In a table of values:
- The ratio between consecutive \( y \)-values is constant.
- This constant ratio indicates an exponential relationship.

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Summary of Key Differences

| Feature | Linear | Quadratic | Exponential |
|------------------|-----------------------|------------------------|-----------------------|
| Equation | \( y = mx + b \) | \( y = ax^2 + bx + c \)| \( y = a(b)^x + c \) |
| Graph Shape | Straight line | Parabola (U-shaped) | Curved line |
| Table Pattern| Constant first diff. | Constant second diff. | Constant ratio |

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Final Answer

The table provides a comprehensive comparison of linear, quadratic, and exponential functions based on their equations, graphs, and patterns in tables. This allows for easy identification of the type of function given its equation, graph, or data points.

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