Problem 18 - Elementary and Intermediate Algebra

Question

To write the slope-intercept form \(y=m x+b\) and the point-slope form \(y-y_{1}=m\left(x-x_{1}\right)\) using function notation, we simply replace \(y\) with \(\square\).

Step-by-Step Solution

Answer

Use \( f(x) = mx + b \) and \( f(x) - y_1 = m(x - x_1) \) for function notation.

Step 1: Understanding Function Notation

In function notation, we often use symbols like \( f(x) \), \( g(x) \), or \( h(x) \) to represent a function. When replacing \( y \) with a function, we use this notation.

Step 2: Replace y with Function Notation in Slope-Intercept Form

In the slope-intercept form \( y = mx + b \), we replace \( y \) with \( f(x) \). Thus, the equation becomes \( f(x) = mx + b \). Here, \( f(x) \) represents the output of the function for any input \( x \).

Step 3: Replace y with Function Notation in Point-Slope Form

In the point-slope form \( y - y_1 = m(x - x_1) \), we also replace \( y \) with \( f(x) \). This results in the equation \( f(x) - y_1 = m(x - x_1) \), where \( f(x) \) represents the function value at \( x \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the simplest ways to express a line mathematically. It is written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept. The slope \( m \) tells us how steep the line is. If \( m \) is positive, the line goes upwards as you move from left to right. A negative \( m \) means the line slopes downwards. The y-intercept \( b \) tells us where the line crosses the y-axis.

This form is particularly useful because it makes it easy to plot the graph of the equation and to understand how changes in \( m \) and \( b \) affect the position and incline of the line. For example:

• If \( m = 2 \) and \( b = 3 \), the equation \( y = 2x + 3 \) represents a line that goes up 2 units for every 1 unit you move right, intersecting the y-axis at 3.

When using function notation, we can write this as \( f(x) = mx + b \), which emphasizes that \( f(x) \) is a function of \( x \). This means for every value of \( x \), there's a matching output \( f(x) \).

Point-Slope Form

The point-slope form of a linear equation is another way to express lines, especially useful when you know a point on the line and the slope. It is written as \( y - y_1 = m(x - x_1) \). Here, \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope. This form highlights the property that a line can be easily drawn if you manage to lock one point and the slope.

Let's imagine you know a point \((3, 4)\) and the slope \(2\). Substituting these into the point-slope form gives \( y - 4 = 2(x - 3) \). This equation shows that if you start at \((3, 4)\), the line moves up 2 units for every 1 unit you move to the right from the point.

In function notation, the same equation is expressed as \( f(x) - y_1 = m(x - x_1) \). This notation signifies that \( f(x) \) is the function value or the y-coordinate at a given \( x \), tying it more clearly to its functional relationship.

Linear Equations

Linear equations are fundamental tools in mathematics. They describe a straight-line relationship between two variables. In the most common forms: the slope-intercept form \( y = mx + b \) and the point-slope form \( y - y_1 = m(x - x_1) \), both portray direct dependencies between the variables \( x \) and \( y \).

Linear equations possess several key characteristics:

• They graph to a straight line.
• The equation’s highest exponent of \( x \) is 1, indicating a single-degree polynomial.
• They have constant rates of change, which the slope \( m \) effectively communicates.

These equations are integral to algebra because they provide insights into how two variables relate across the coordinate plane. When expressed in function notation like \( f(x) = mx + b \) or \( f(x) - y_1 = m(x - x_1) \), they become useful in a variety of mathematical and scientific disciplines, especially when analyzing real-world situations through models and graphs.