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Chapter 8: Problem 40

Write an equation of each line using function notation. Slope 0 ; through \((-10,23)\)

Short Answer

Expert verified
The function is \( f(x) = 23 \) representing the line.

Step by step solution

01

Understand the Problem

The problem asks us to write the equation of a line in function notation, given the slope is 0 and it passes through the point (-10, 23).
02

Recall Slope of Horizontal Line

When a line has a slope (m) of 0, it is a horizontal line. Therefore, the equation of the line can be expressed as a constant function \( f(x) = b \).
03

Determine the Constant

Since the line is horizontal and passes through the point (-10, 23), the y-coordinate of this point determines the constant value of the line. Hence, the function should be \( f(x) = 23 \).
04

Write the Function in Notation

The equation of the line, in function notation, is \( f(x) = 23 \). This represents a function where every input x gives an output of 23.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope of a Line
The slope of a line is a measure of its steepness and direction. It's usually represented by the letter \( m \).
For example:
  • A positive slope means the line rises as you move from left to right.
  • A negative slope means the line falls as you move from left to right.
  • A zero slope means the line is perfectly horizontal, with no steepness at all.
To compute the slope of a non-horizontal or non-vertical line between two points \((x_1, y_1)\) and \((x_2, y_2)\), the formula is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]If \( m = 0 \), it means there is no rise or fall, indicating it's a horizontal line. Understanding this concept is key when dealing with linear equations and graphing.
Horizontal Line Equation
A horizontal line has a unique characteristic: its slope is always 0. This kind of line runs parallel to the x-axis.
In mathematical terms, the equation of a horizontal line can be expressed as:\[y = b\]Here, \( b \) is a constant, which represents the y-value where the line sits.
All points on this line share this same y-value, even as x changes. For example, if a horizontal line passes through the point \((-10, 23)\), the entire line will lie at \( y = 23 \).Some key points to remember:
  • The x-value doesn't affect the y-value.
  • A horizontal line equation does not contain the variable \( x \).
  • Every point on the line can be represented with the constant \( y \) value.
This makes it simple to write the equation for any horizontal line once you have a point it passes through.
Function Notation
Function notation provides a powerful way to represent functions mathematically. In basic terms, a function like \( f(x) \) is a rule that assigns to each value of \( x \) exactly one value of \( f(x) \).
It's a concise way to write the relationship between two variables, typically allowing easy substitution and evaluation. Key aspects of function notation include:
  • \( f \) indicates the name of the function.
  • \( x \) is the input or independent variable.
  • \( f(x) \) is the output or the dependent variable.
Moreover, if we have a horizontal line with a slope of 0, the function can be expressed as \( f(x) = b \), where \( b \) is constant.For example, for a horizontal line through the point \((-10, 23)\), the function notation is \( f(x) = 23 \). This means regardless of the \( x \) value, \( f(x) \) will always output 23.

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