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Chapter 8: Problem 19

Find an equation of the line with the given slope and containing the given point. Write the equation using function notation. Slope 3 ; through \((1,2)\)

Short Answer

Expert verified
The equation is \( f(x) = 3x - 1 \).

Step by step solution

01

Identify the Slope-Point Form of a Line

To find a line equation with a given slope and point, use the point-slope form: \( y - y_1 = m(x - x_1) \). Here, \( m \) is the slope, and \((x_1, y_1)\) is the point through which the line passes.
02

Substitute the Given Values

Plug the given slope and point into the point-slope formula. The slope \( m \) is 3 and the point is \((1,2)\). Substitute: \( y - 2 = 3(x - 1) \).
03

Simplify the Equation

Distribute the slope 3 on the right-hand side and simplify: \( y - 2 = 3x - 3 \).
04

Solve for y

Add 2 to both sides of the equation to isolate \( y \): \( y = 3x - 1 \).
05

Write in Function Notation

Express the equation in function notation: \( f(x) = 3x - 1 \). This shows that the function \( f \) which outputs \( y \) depends on \( x \).

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Key Concepts

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

Point-Slope Form
The point-slope form is a convenient way of writing the equation for a line when you know a point on the line and the slope. The point-slope form equation is: \(y - y_1 = m(x - x_1)\), where:
  • \(m\) represents the slope of the line.
  • \((x_1, y_1)\) is a specific point on the line.
This method simplifies finding the equation of a line without needing information about the y-intercept upfront. It provides a direct route to establishing the line鈥檚 relationship between the point and its slope. To use it, you simply need to substitute the values of the slope and the known point into the equation. This makes it very useful for quickly determining the linear equation of a line when these two key components are known.
Slope
The slope of a line is a measure of its steepness and direction. In a linear equation, it is denoted by \(m\) and can be calculated using two points from the line: \((x_1, y_1)\) and \((x_2, y_2)\). The formula for slope is: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here鈥檚 how slope works:
  • A positive slope means the line inclines upwards as it moves from left to right.
  • A negative slope indicates the line declines downwards.
  • A zero slope means the line is horizontal, showing no vertical change.
  • An undefined slope signifies a vertical line where the x-values are the same.
Understanding slope is crucial for graphing lines and predicting their behavior based on changes in \(x\). It lays the foundation for delving deeper into coordinate geometry and understanding how changes in one variable affect the other in linear relationships.
Function Notation
Function notation is a way of defining a function in terms of an input-output relationship. It simplifies equations by replacing \(y\) with \(f(x)\), indicating the relationship between \(x\) and its output. Function notation is expressed as \(f(x) = mx + b\), where:
  • \(f(x)\) is the output, also known as the dependent variable.
  • \(x\) is the input or independent variable.
  • \(m\) and \(b\) are constants, where \(m\) is the slope and \(b\) is the y-intercept.
Using function notation helps to express the function clearly and allows for easy graphing and evaluation at specific points. It communicates that the output is derived from input \(x\) through a defined rule or formula. This notation keeps the equation concise and emphasizes its nature as a function, which is especially useful in more advanced mathematics.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, involves using algebra to describe and understand geometry. It uses a coordinate system, typically the Cartesian plane, to analyze shapes and solve geometric problems.
  • Points on the plane are represented as \((x, y)\) coordinates.
  • It helps in defining line equations, such as the one obtained from the point-slope form.
  • Slope is a key concept in coordinate geometry, as it is instrumental in understanding line characteristics.
By using coordinate geometry, one can visually and algebraically understand the elements of algebraic expressions. It bridges the gap between algebra and geometry, making it easier to solve complex problems by providing a visual interpretation of mathematical theories.

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Most popular questions from this chapter

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