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Magazine Chapter 8: Problem 46
Identify the type of function represented by each equation. Then graph the equation. (lesson 8.5\()\) $$ y=0.8 x $$
Short Answer
Expert verified
The function is linear with a slope of 0.8 and passes through the origin.
Step by step solution
01
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 \( c = 0 \).
02
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).
03
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.
04
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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:
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 \( x \).
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:
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.
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:
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.
