Problem 40 - Algebra Form and Function

Question

Give the slope and \(y\) -intercept for the graphs of the functions. $$ f(x)=\frac{x}{7}-12 $$

Step-by-Step Solution

Answer

Answer: The slope of the function is \(\frac{1}{7}\) and the y-intercept is \(-12\).

Step 1: Identify the slope

The slope of a linear function in the form \(f(x) = mx + b\) is represented by the coefficient of \(x\). In the given function, \(f(x) = \frac{1}{7}x - 12\), the coefficient of \(x\) is \(\frac{1}{7}\). Therefore, the slope is \(\frac{1}{7}\).

Step 2: Identify the y-intercept

The y-intercept of a linear function in the form \(f(x) = mx + b\) is represented by the constant term, \(b\). In the given function, \(f(x) = \frac{1}{7}x - 12\), the constant term is \(-12\). Therefore, the y-intercept is \(-12\).

Key Concepts

Slope of a Line

The slope of a line in a linear function is crucial because it describes how steep the line is. Imagine you are thinking about a hill. The slope tells you how steep this hill is. For linear equations, the slope is represented by the coefficient of the variable \(x\). It is often denoted as \(m\).
The slope can be understood as the "rise over run," which is the ratio of the vertical change to the horizontal change between two distinct points on a line. A positive slope means the line is rising, or going uphill as you move from left to right. Conversely, a negative slope means the line is going downhill.
In our function \(f(x) = \frac{1}{7}x - 12\), the slope is \(\frac{1}{7}\). This tells us that the line increases by \(\frac{1}{7}\) units in the vertical direction for every 1 unit it moves in the horizontal direction. The slope is quite gentle, given the small value \(\frac{1}{7}\).

• The slope is \(\frac{1}{7}\), which is positive, indicating the line is slightly rising.
• Remember, when the slope is zero, the line is perfectly horizontal.
• If the slope were negative, the line would descend from left to right.

Y-Intercept

The \(y\)-intercept of a linear function is the value where the line cuts through the \(y\)-axis. Simply, it is the point on the graph where \(x\) equals zero. It's like the starting point of the line when you view it on a graph.
For any linear equation in the form \(f(x) = mx + b\), the \(y\)-intercept is the constant term \(b\). This is important because it provides a fixed point for drawing the line.
In the function \(f(x) = \frac{1}{7}x - 12\), the \(y\)-intercept is \(-12\). This means that the line will cross the \(y\)-axis at \(-12\).

• The \(y\)-intercept is always written as \((0, b)\). Hence, here it is \((0, -12)\).
• It informs us where the line begins on the \(y\)-axis when \(x = 0\).

By understanding the \(y\)-intercept, you can more easily sketch the line as it gives you a key starting point on the graph.

Linear Equations

Linear equations are mathematical expressions that represent straight lines when plotted on a graph. They have a standard form, \(ax + by = c\), or more commonly, \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept.
One of the simplest ways to understand a linear equation is to visualize it on a two-dimensional plane. Each point on the line is a solution to the equation.
In our example, \(f(x) = \frac{1}{7}x - 12\), we can see exactly how the components of this equation form a line:

• The term \(\frac{1}{7}x\) indicates how the line is slanted with its slope.
• The term \(-12\) gives us the place where this line will intercept the \(y\) axis.

This type of equation is linear because when graphed, every value of \(x\) corresponds to one value of \(y\), creating a straight line.