Problem 46 - Algebra 2

Question

Identify the type of function represented by each equation. Then graph the equation. (lesson 8.5\()\) $$ y=0.8 x $$

Step-by-Step Solution

Answer

The function is linear with a slope of 0.8 and passes through the origin.

Step 1: Identify the type of function

The given equation is in the form of \( y = mx \), where \( m = 0.8 \). This is a linear equation because it is in the form \( y = mx + c \), where \( c = 0 \). The graph of such an equation is a straight line that passes through the origin, (0,0), given that \( c = 0 \).

Step 2: Determine the slope and y-intercept

In the equation \( y = 0.8x \), the slope \( m \) is 0.8, and the y-intercept \( c \) is 0. Using this information, we know that for every 1 unit increase in \( x \), \( y \) increases by 0.8 units, and the line crosses the y-axis at (0,0).

Step 3: Plot points for the graph

To graph \( y = 0.8x \), select values for \( x \) and calculate the corresponding \( y \) values. For example, if \( x = 1 \), then \( y = 0.8 \). If \( x = 2 \), then \( y = 1.6 \). Plot these points on the coordinate plane, including the point (0,0) for the y-intercept.

Step 4: Draw the line

Once you have plotted several points, draw a straight line through these points, extending through both quadrants. The line should pass through the origin and reflect the consistent slope of 0.8.

Key Concepts

Understanding Slope in Linear Equations

In linear equations, the slope is a critical concept that helps us understand how a line behaves on a graph. The slope, often denoted by \( m \), represents the rate of change of the line. It tells us how much \( y \) changes for a unit increase in \( x \).
Essentially, the slope answers the question: "How steep is the line?". A positive slope like 0.8 means the line rises as it moves from left to right.
Here are some important points about slope:

• A larger positive slope results in a steeper upward line.
• A negative slope indicates that the line goes downward.
• A slope of zero represents a horizontal line.
• An undefined slope (like in vertical lines) means the line goes straight up and down.

In our equation \( y = 0.8x \), the slope tells us that for every step rightward on the x-axis, the line moves up 0.8 units. This consistent rate of change is what defines a linear function's path.

Y-Intercept: Defining Line's Starting Point

The y-intercept is another fundamental component of a linear equation. This is the point where the line crosses the y-axis, and it is represented by \( c \) in the equation \( y = mx + c \).
When the y-intercept is zero, as in \( y = 0.8x \), the line begins at the origin (0,0) on the graph.
Key features of the y-intercept include:

• It indicates where the line starts when \( x = 0 \).
• If \( c \) is positive, the line crosses above the origin; if negative, below the origin.
• In \( y = mx \), the y-intercept is always zero, signaling a direct proportionality between \( y \) and \( x \).

Understanding the y-intercept helps us efficiently graph linear equations, allowing us to draw the line where it naturally begins on the coordinate plane.

Graphing Linear Functions

Graphing linear functions is an intuitive way to visualize relationships between variables. When you graph a linear function like \( y = 0.8x \), you can see how changes in \( x \) affect \( y \).
Here are steps to successfully graph a linear function:

• Calculate the y-values by plugging x-values into the equation. For example, \( x = 1 \) gives \( y = 0.8 \).
• Plot these calculated points on a coordinate plane. Start with easy values like (0,0).
• Draw a line through all the plotted points. Ensure the line is straight and extends in both directions.

Once the line is drawn, the slope and y-intercept become visually clear. This makes it easier to make predictions about \( y \) for unknown \( x \) values by simply following the direction of the line.

Navigating the Coordinate Plane

The coordinate plane is a two-dimensional grid that helps us plot and interpret the behavior of functions like linear equations. It's made up of two axes: the horizontal x-axis and the vertical y-axis.
Understanding the coordinate plane is crucial for graphing and analyzing linear equations. Here's what you need to know:

• The origin is the center point where both axes meet, at (0,0).
• Quadrants are created by the axes, each having positive or negative values.
• Lines are plotted based on pairs of \( (x, y) \) coordinates derived from the function.

Using the coordinate plane allows us to transform abstract equations into tangible graphs. When graphed, a linear equation like \( y = 0.8x \) becomes a visible line, helping us better understand the mathematical relationship it describes.