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Chapter 0: Problem 16

Graph. List the slope and y-intercept. $$ g(x)=-x+3 $$

Short Answer

Expert verified
The slope is \(-1\) and the y-intercept is \(3\).

Step by step solution

01

Identify the equation format

The given equation is in the format of a linear equation, \( g(x) = -x + 3 \), which can also be written as \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.
02

Determine the slope

In the equation \( y = -x + 3 \), compare this with \( y = mx + b \). The coefficient of \( x \) is \(-1\), which means the slope \( m \) is \(-1\).
03

Determine the y-intercept

In the same equation \( y = -x + 3 \), the constant term is \( 3 \). This means the y-intercept \( b \) is \( 3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Slope
The slope of a line in a linear equation is a key concept when graphing or interpreting data. In the equation form \( y = mx + b \), the slope is represented by \( m \). The slope describes the steepness and direction of the line. A positive slope means the line is rising, moving upwards as it goes from left to right. Conversely, a negative slope means the line is falling, sloping downwards as it moves from left to right. In our specific example, \( g(x) = -x + 3 \), the slope \( m \) is \(-1\).
  • A slope of \(-1\) means for each unit you move horizontally to the right, you move one unit down vertically.
  • The slope is a ratio of the 'rise' over the 'run'. In this case, the 'rise' is \(-1\) unit, and the 'run' is \(1\) unit.
Understanding slope is crucial because it helps you quickly ascertain how a line behaves over the rectangular coordinate system.
Exploring the Y-Intercept
The y-intercept in a linear equation indicates where the line crosses the y-axis. This is the point on the graph where the x-value is zero. In the equation \( y = mx + b \), \( b \) represents the y-intercept.For our equation \( g(x) = -x + 3 \), the y-intercept \( b \) is \(3\). It illustrates that the line will touch the y-axis at the point (0, 3).
  • The y-intercept is particularly useful as it provides a starting point for drawing the line on a graph.
  • It's also a critical component in figuring out how the graph relates to real-life situations where initial conditions or values are known.
Always remember, identifying the y-intercept assists in both plotting the graph and understanding the boundary conditions of an equation.
Graphing a Linear Equation
Graphing a line involves utilizing both the slope and the y-intercept of the equation. To graph the line represented by \( g(x) = -x + 3 \), start by plotting the y-intercept.
  • Place a point on the graph where \( y = 3 \) and \( x = 0 \). This is the y-intercept.
  • From this point, use the slope to place the next point. Since the slope is \(-1\), you move down one unit vertically and one unit to the right horizontally.
Once you have at least two points, you can draw a straight line through them extending in both directions. This line is the graphical representation of the equation \( g(x) = -x + 3 \).
  • The graph pictorially demonstrates the relationship between independent variable \( x \) and dependent variable \( y \).
  • Understanding this relationship helps in analyzing and solving real-world problems.
Drawing graphs manually or using technology can enhance comprehension and interpretation of linear relationships.

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