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 \( f(x) \).
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.
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.
I hope this helps clarify these important concepts in mathematics!
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