Problem 43 - Algebra Form and Function

Question

Give the slope and \(y\) -intercept for the graphs of the functions. $$ f(x)=\pi x $$

Step-by-Step Solution

Answer

Answer: The slope of the function is \(\pi\), and the y-intercept is 0.

Step 1: Write the function in slope-intercept form (y = mx + b)

Here, the given function is already in the slope-intercept form: $$ f(x)=\pi x $$ So, we can write it as: $$ y=\pi x $$

Step 2: Identify the slope (m)

In the equation \(y=\pi x\), the coefficient of x is the slope (m). So the slope is: $$ m=\pi $$

Step 3: Identify the y-intercept (b)

In the equation \(y=\pi x\), there is no constant term (b). That means the y-intercept is 0. So we have: $$ b=0 $$

Step 4: The slope and y-intercept of the function

The slope (m) of the function \(f(x)=\pi x\) is \(\pi\), and the y-intercept (b) is 0.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing linear functions and equations. It is written in the format: \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept. This form is particularly useful because it clearly shows both the steepness of the line and where it crosses the y-axis.
The equation of a line directly tells us these two important characteristics: the slope and the intercept. Let's break down each part:

• The **slope** \( (m) \) tells us how steep the line is and the direction it goes. A larger absolute value of \( m \) means a steeper slope, while a smaller value indicates a gentler slope.
• The **y-intercept** \( (b) \) is the value where the line crosses the y-axis. It tells us what the output (or \( y \)-value) is when the input \( x \) is zero.

By rearranging or identifying parts of an equation in this form, we can easily sketch graphs and understand the behavior of linear functions.

Slope

The slope of a line is a value that shows the rate at which \( y \) changes with respect to \( x \). It describes the rise over run, or vertical change over horizontal change, between any two points on a line.
In the slope-intercept form equation \( y = mx + b \):

• **\( m \)** is the slope. For example, in the equation \( y = \pi x \), the slope \( m \) is \( \pi \). This means that for every one-unit increase in \( x \), \( y \) increases by \( \pi \) units.

To decide whether a slope is positive or negative, observe its sign. A positive slope means the line goes upwards from left to right, and a negative slope means it goes downwards. Understanding the slope helps in predicting how changes in one variable affect the other, which is essential in many fields like science, economics, and engineering.

Y-Intercept

The y-intercept is where the line crosses the y-axis on the graph. It is an essential coordinate because it gives us a specific point through which the line passes. In the slope-intercept form equation \( y = mx + b \), the **y-intercept** is represented by \( b \).
For example, in the equation \( y = \pi x \), there is no \( b \) term present, which means \( b = 0 \). Thus, the y-intercept is 0. This indicates that the line intersects the y-axis at the origin point \((0, 0)\).
Understanding the y-intercept helps when you need to graph a function quickly or when you're trying to understand starting conditions, like knowing the initial amount of a changing quantity. If there is no constant or visible intercept term in the function, it's default to assume the y-intercept is zero, significantly simplifying certain calculations and interpretations.