Question

Vocabulary Terms - Define/Illustrate

Function Notation
Vertical Line test
Independent vs Dependent variable

Answer 1

Sure! Here are definitions and illustrations for the listed vocabulary terms:

### Function Notation
**Definition**: Function notation is a way to represent functions in mathematics. It’s typically expressed as \( f(x) \), where \( f \) denotes the function, and \( x \) is the input variable. The function takes \( x \) as input and produces an output.

**Illustration**:
For example, if we have a function \( f(x) = 2x + 3 \), we can compute the output for specific inputs:
- If \( x = 1 \), then \( f(1) = 2(1) + 3 = 5 \).
- If \( x = 2 \), then \( f(2) = 2(2) + 3 = 7 \).

In this example, we can evaluate the function for various values of \( x \) using the notation \( f(x) \).

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### Vertical Line Test
**Definition**: The vertical line test is a method used to determine if a graph represents a function. According to this test, if any vertical line drawn on the graph crosses the graph at more than one point, then the graph does not represent a function.

**Illustration**:
Imagine a graph showing a circle:
- If you draw a vertical line through the circle, it will intersect the circle at two points, indicating that for some values of \( x \), there are multiple \( y \) values (e.g., the points where the line intersects the circle).

In contrast, a graph of \( y = x^2 \) (a parabola) would pass the vertical line test since any vertical line drawn will intersect the parabola at only one point.

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### Independent vs Dependent Variable
**Definition**:
- **Independent Variable**: The variable that is manipulated or changed in an experiment or function. It is usually plotted on the x-axis.
- **Dependent Variable**: The variable that is observed or measured in response to changes in the independent variable. It is usually plotted on the y-axis.

**Illustration**:
Consider the function \( y = 3x + 2 \):
- **Independent Variable** \( (x) \): This is the value you can control or change. You could set \( x \) to any number.
- **Dependent Variable** \( (y) \): This is the value that depends on \( x \). When you change \( x \), the value of \( y \) changes accordingly.

If you let \( x = 1 \), then:
- \( y = 3(1) + 2 = 5 \). So, when \( x \) changes, \( y \) changes as well.

In a real-world scenario, consider the relationship between hours studied (independent variable) and test scores (dependent variable): Increasing the study hours may lead to higher test scores.

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I hope this helps clarify these important concepts in mathematics!