Problem 49 - Algebra 1

Question

Find the slope of the graph of the linear function \(f\). $$ f(-1)=2, f(3)=2 $$

Step-by-Step Solution

Answer

The slope of the graph of the given linear function is 0.

Step 1: Identify Known Values

The function f's values are given for two specific x coordinates: f(-1) = 2 and f(3) = 2. Here, -1 and 3 are the x-coordinates \(x_1\) and \(x_2\), respectively, and 2 and 2 are the corresponding y-coordinates \(y_1\) and \(y_2\), respectively.

Step 2: Substitute Values into Slope Equation

Using the slope equation \(m = (y_2 - y_1)/(x_2 - x_1)\), you insert the known x and y values. This gives \(m = (2 - 2)/(3 - -1)\).

Step 3: Simplify the Equation to Find the Slope

Simplifying the denominator, you get \(m = (2 - 2)/(3 + 1)\). Further simplifying both the numerator and the denominator, the equation transforms into \(m = (0)/(4)\). Obtaining the final result, \(m = 0\). Hence, the slope is 0.

Key Concepts

Linear Function

A linear function is one of the simplest types of functions you can encounter in mathematics. It is defined by an equation of the form \( f(x) = mx + b \). Here, \( m \) is the slope of the line, and \( b \) is the y-intercept. The concept of linear functions can be grasped through a couple of key characteristics.

• Proportionality: In a linear function, each change in \( x \) translates to a consistent change in \( f(x) \). This means the relationship between \( x \) and \( f(x) \) is direct and proportional.
• Graph is a Straight Line: Due to its linearity, the graph of this function will always be a straight line.

Understanding the properties of a linear function is crucial when studying mathematics, especially when you need to graph the function or find its slope and intercept.

Slope Formula

The slope formula is a fundamental tool used to measure the steepness and direction of a line. The slope, often represented with the letter \( m \), is calculated using the difference in y-coordinates divided by the difference in x-coordinates of two points on the line. The formula is expressed as:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

• Positive Slope: If \( m > 0 \), the line increases as it moves from left to right.
• Negative Slope: If \( m < 0 \), the line decreases as it moves from left to right.
• Zero Slope: If \( m = 0 \), the line is perfectly horizontal, showing no change in y-value as x changes.
• Undefined Slope: If the denominator \( x_2 - x_1 \) equals zero, the slope is undefined, indicating a vertical line.

In the original exercise, when the values \( f(-1) = 2 \) and \( f(3) = 2 \) were identified, substituting into the slope formula resulted in a zero slope, confirming the graph of this function is horizontal.

Graph of a Function

The graph of a function is a visual representation of all the output values (y-values) corresponding to the input values (x-values) for a function. In the case of linear functions, this graph is characterized by a straight line, whose nature is significantly influenced by its slope.

• Horizontal Line: When the slope of a linear function is 0, the graph is a horizontal line. This indicates all y-values are the same across different x-values, as in the given function \( f(x) = 2 \) for \( x = -1 \) to \( x = 3 \).
• Vertical Line: With an undefined slope, a graph becomes a vertical line, usually impossible for linear functions expressed with an equation \( f(x) = mx + b \) because it would imply a division by zero.
• Slope Effects: The steeper the slope, the more dramatic the rise or fall of the line. A flatter slope indicates a gentler incline or decline.

For students, the graph serves as a unique tool to understand and visualize how changes in coefficients like the slope \( m \) and intercept \( b \) affect the overall shape and position of the line on a coordinate plane.