Problem 44 - Algebra 1

Question

Find the slope of the graph of the linear function \(f\). $$ f(-3)=-9, f(3)=9 $$

Step-by-Step Solution

Answer

The slope of the graph of the linear function \(f\) is 3.

Step 1: Identify Coordinates

Identify two points on the line from the function inputs and outputs. This can be done by taking the inputs as x-coordinates and the corresponding outputs as y-coordinates. Thus, (-3, -9) and (3, 9) are the identified points.

Step 2: Apply Formula

Apply the formula \(m = \frac{(y2 - y1)}{(x2 - x1)}\) using the two points obtained in step one. In this case, substituting, we have: \(m = \frac{(9 - (-9))}{(3 - (-3))}\).

Step 3: Simplify the Expression

Simplify the expression obtained in step two to find the slope of the line. For the current exercise, this simplifies to \(m = \frac{18}{6}\).

Step 4: Calculate the Result

Carry out the division to determine the final value of the slope. Thus, we have the slope \(m = 3\).

Key Concepts

Linear Equation

A linear equation represents a straight line when plotted on a graph. It is the most basic of algebraic expressions and is key to understanding coordinate geometry and calculating slope. Formally, a linear equation can be written in the form: \[ y = mx + b \] where:

• \( y \) is the dependent variable
• \( m \) is the slope
• \( x \) is the independent variable
• \( b \) is the y-intercept, the point where the line crosses the y-axis

Linear equations do not involve exponents higher than one, and they graph as a simple straight line. This line shows a constant rate of change, which is essential when analyzing data or trends in areas like economics or physics. The slope \( m \) is constant, meaning the rate of change between any two points on the graph remains the same.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe the position of points, lines, and shapes on a plane. Using coordinates, we can precisely define points and use numerical methods to find relationships between them. Every point on the plane is identified by an ordered pair of numbers, \((x, y)\), which describe its location in relation to two perpendicular axes:

• The x-axis, which runs horizontally
• The y-axis, which runs vertically

In the context of our linear function, points like \((-3, -9)\) and \((3, 9)\) are specified in this coordinate space. By employing these coordinates, we can calculate the slope and analyze the properties of the line that they constitute. Coordinate geometry is particularly useful because it allows us to visualize algebraic equations and solve geometric problems using algebraic techniques.

Calculating Slope

The slope of a line tells us how steep the line is and the direction it is going. It is one of the fundamental concepts in coordinate geometry. The slope is calculated using the formula:\[ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \]where:

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of two distinct points on the line
• \( m \) represents the slope

The slope \( m \) describes the change in \( y \) for every unit change in \( x \). A positive slope means the line rises from left to right, while a negative slope means it falls. In the case of our example, the calculated slope of 3 indicates a steep upward incline as you move from left to right. Calculating slope is vital for understanding linear relationships in data, as it represents the rate of change between variables.