Problem 17 - Algebra 2

Question

Identify the type of function represented by each equation. Then graph the equation. \(y=2.5 x\)

Step-by-Step Solution

Answer

The function is linear, represented by a straight line with a slope of 2.5.

Step 1: Identify the Type of Function

The given equation is in the form of \(y = mx\), where \(m\) is a constant. This is the format for a linear equation, which represents a straight line when graphed. In this equation, \(m = 2.5\), thus it's a linear function.

Step 2: Determine Key Characteristics

For linear functions of the form \(y = mx\), the slope \(m\) can be identified directly from the equation. Here, the slope is 2.5, indicating the line rises by 2.5 units for every 1 unit it moves to the right.

Step 3: Plot the Graph

Start by plotting the origin point \((0,0)\) as it satisfies the equation \(y = 2.5x\). Next, use the slope to find another point: if \(x = 1\), then \(y = 2.5 \times 1 = 2.5\). Plot this point \((1, 2.5)\). Connect these points with a straight line extending in both directions.

Key Concepts

Linear Equations

Linear equations are a foundational concept in algebra and mathematics. They represent relationships between variables where the change in one variable is consistent with the change in another. A simple linear equation is generally in the form of \(y = mx + b\), where:

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

In the case of the exercise, the equation \(y = 2.5x\) omits the \(b\) term, which simplifies to \(b = 0\). This means the line passes through the origin \((0, 0)\). Linear equations like these are crucial because they model real-world situations where a consistent rate of change applies, such as speed, distance, and cost analysis.

Graphing Linear Equations

Graphing linear equations involves plotting points on a graph that represent solutions to the equation. It's a visual representation that helps in understanding the behavior of equations.
To graph the equation \(y = 2.5x\):

• Begin by identifying a starting point, often the y-intercept. Here, it's the origin \((0, 0)\) since the graph of \(y = mx\) goes through the origin.
• Next, use the slope \(m\). A slope of 2.5 means you go up 2.5 units in the vertical direction for every 1 unit you move in the horizontal direction. This gives a rise/run ratio.
• From the origin, count 1 unit to the right and 2.5 units up to find another point, \((1, 2.5)\).
• Plot this second point and draw a straight line through both points. Extend this line in both directions.

Graphing allows students to visualize how changes in the equation will affect the line, helping to solve problems related to intercepts and intersections with other lines.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so that you can easily identify the slope and y-intercept of the graph. This form is given as \(y = mx + b\), which translates to:

• \(m\), the slope, indicates the steepness or incline of the line. A positive \(m\) means the line rises as it moves from left to right, while a negative \(m\) means it falls.
• \(b\), the y-intercept, is the value of \(y\) when \(x = 0\). It shows where the line crosses the y-axis.

The equation \(y = 2.5x\) is an example of a linear equation in slope-intercept form where \(b = 0\). Thus, the line intersects the y-axis at the origin \((0,0)\). Understanding this form helps in quickly translating algebraic expressions into graphical representations, making it easier to see and analyze relationships between quantities on a graph.