/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Graph each linear function. $$... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 1

Graph each linear function. $$ f(x)=-2 x $$

Short Answer

Expert verified
Plot points at \((0,0)\) and \((1,-2)\) to graph the line.

Step by step solution

01

Identify the Function

The function provided is \( f(x) = -2x \). This is a linear function where the coefficient of \( x \) is \(-2\), which represents the slope, and there is no constant term, meaning the y-intercept is \(0\).
02

Determine the Slope and Y-Intercept

In the equation \( y = mx + b \), \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = -2 \) and \( b = 0 \). The slope of \(-2\) indicates that for every unit increase in \( x \), \( y \) decreases by 2 units.
03

Plot the Y-Intercept

Start plotting the graph by marking the y-intercept at \( (0,0) \). This is the point where the graph crosses the y-axis.
04

Use the Slope to Find Another Point

From the y-intercept \((0,0)\), use the slope \(-2\) to find another point. Move 1 unit to the right (positive x-direction) and 2 units down (negative y-direction) to arrive at the point \((1, -2)\).
05

Plot the Second Point and Draw the Line

Plot the point \((1, -2)\) on the graph. Then, draw a straight line through both points \((0,0)\) and \((1, -2)\) to complete the graph of the linear function.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is an essential way to express a linear function. It is written as \( y = mx + b \), where:
  • \( m \) represents the slope of the line.
  • \( b \) signifies the y-intercept, the point where the line crosses the y-axis.
This form makes it easy to identify both the slope and the y-intercept directly from the equation. In the example function \( f(x) = -2x \), it can be rewritten in slope-intercept form as \( y = -2x + 0 \). Here, \( m = -2 \) and \( b = 0 \). This tells us that the line has a slope of \(-2\) and crosses the y-axis at \(0\). Understanding this form allows you to quickly sketch the graph of any linear function.
Y-Intercept
The y-intercept is a critical concept when graphing linear functions. It is the point where the line crosses the y-axis of the graph.
This occurs when \( x = 0 \). In the slope-intercept form \( y = mx + b \), \( b \) represents the y-intercept.
For the function \( f(x) = -2x \), the y-intercept \( b \) is \(0\).
  • This means the graph will pass through the origin, the point \((0,0)\).
  • Identifying the y-intercept is the first step in plotting the graph of a linear function.
By starting with this point, you can use the slope to determine the direction and steepness of the line.
Plotting Points
Plotting points is a straightforward yet vital step in graphing linear functions. Once the y-intercept is established, the slope guides the placement of additional points.
The slope, defined as \( m \), tells you how to move from one point to another on the graph.
  • In our example, the slope \(-2\) means that for every 1 unit you move to the right along the x-axis, you should move 2 units down in the y-direction.
  • Starting from the y-intercept \((0,0)\), this rule takes you to the next point, \((1, -2)\).
After identifying this second point, plot it on the graph. Connect the dots with a straight line and you have successfully graphed the linear function. This method ensures accuracy in depicting the intended line on a graph.

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Most popular questions from this chapter

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