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Chapter 8: Problem 68

Sketch the graph of each function. See Section 8.5. $$ h(x)=x-3 $$

Short Answer

Expert verified
The graph of \( h(x) = x - 3 \) is a straight line with slope 1 and y-intercept -3.

Step by step solution

01

Identify the Type of Function

The function given is a linear function because it is of the form \( h(x) = mx + b \), where \( m = 1 \) and \( b = -3 \). Linear functions graph as straight lines.
02

Determine the Y-intercept

The y-intercept of the function can be determined by setting \( x = 0 \). Calculate \( h(0) = 0 - 3 = -3 \). Therefore, the y-intercept is \( -3 \). This tells us that the graph crosses the y-axis at \( (0, -3) \).
03

Determine the Slope

The slope \( m \) of the function is \( 1 \), indicating that for every unit increase in \( x \), \( h(x) \) increases by 1 unit. This shows the steepness of the line.
04

Plot the Y-intercept

To begin, plot the point \( (0, -3) \) on the graph. This is the starting point of the line.
05

Use the Slope to Plot Another Point

Using the slope \( m = 1 \), find another point. From \( (0, -3) \), move 1 unit up and 1 unit right to plot the next point at \( (1, -2) \).
06

Draw the Line

Connect the plotted points \( (0, -3) \) and \( (1, -2) \) with a straight line. Extend the line in both directions, which represents the function \( h(x) = x - 3 \).

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Key Concepts

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

Graphing Linear Equations
Graphing linear equations is a fundamental skill in algebra. A linear equation describes a straight line when graphed on a coordinate plane. These equations are typically written in the form \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept.
To graph a linear equation:
  • Start by identifying the y-intercept, which is where the line crosses the y-axis.
  • Next, use the slope to determine the direction and steepness of the line.
  • Plot the y-intercept on the graph, and use the slope to find other points on the line.
  • Draw a straight line through the plotted points.
Graphing these equations helps visually understand how changes in the slope and y-intercept affect the line's position and angle on the graph.
Y-Intercept
The y-intercept is a vital part of understanding linear functions. It's the point where the graph of the equation crosses the y-axis. In the equation form \( y = mx + b \), the y-intercept is denoted by \( b \).
To find the y-intercept:
  • Set \( x = 0 \) in the equation.
  • Solve for \( y \).
For example, in the function \( h(x) = x - 3 \), set \( x = 0 \) to find \( h(0) = 0 - 3 = -3 \). Thus, the y-intercept is \( -3 \), which tells us the line crosses the y-axis at the point \( (0, -3) \).
The y-intercept is crucial for graphing because it provides a starting point for the line.
Slope
The slope of a linear equation defines its steepness and direction. It's represented by the coefficient of \( x \) in the equation \( y = mx + b \). The slope \( m \) can be thought of as 'rise over run', indicating how much the function value increases (rise) for a one-unit increase in \( x \) (run).
For instance:
  • If \( m = 1 \), for every increase in \( x \) by 1, \( y \) increases by 1. This means the line rises at a 45-degree angle.
  • Positive slopes indicate an upward trend as you move right, while negative slopes indicate a downward trend.
  • A slope of zero means a horizontal line, while an undefined slope means a vertical line.
Understanding the slope is essential for predicting how the line will behave and how changing variables will affect its trajectory.
Straight Lines
Straight lines are the simplest forms of graphs in mathematics. When you solve a linear equation, the resulting graph is always a straight line. This is because linear equations show direct proportional relationships between variables.
Key characteristics of straight lines include:
  • They have a constant slope, meaning the steepness doesn't change.
  • The equation \( y = mx + b \) always produces a straight line when graphed.
  • Each point on the line solves the equation, showing a consistent relationship between \( x \) and \( y \).
Straight lines are important in algebra as they provide a clear and predictable model of relationships and changes over a given set of inputs. Whether you're analyzing data trends or solving algebraic expressions, understanding how straight lines work is essential.

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