/*! 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 4 Graph each linear function. $$... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 4

Graph each linear function. $$ f(x)=2 x+6 $$

Short Answer

Expert verified
Plot points (0, 6) and (1, 8), then draw a line through them.

Step by step solution

01

Identify the Slope and Y-Intercept

First, recall the slope-intercept form of a linear equation, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In this function, \( f(x) = 2x + 6 \), the slope \( m \) is 2 and the y-intercept \( b \) is 6.
02

Plot the Y-Intercept

Start by plotting the y-intercept on the graph. Here, the y-intercept is 6, so place a point at \( (0, 6) \) on the y-axis. This is the point where the line crosses the y-axis.
03

Use the Slope to Find Another Point

The slope \( m = 2 \) means rise over run = \( \frac{2}{1} \). From the y-intercept \( (0, 6) \), move up 2 units and 1 unit to the right to find the next point, which is \( (1, 8) \). Plot this point on the graph.
04

Draw the Line

Draw a straight line through the two points \( (0, 6) \) and \( (1, 8) \). This line represents the graph of the function \( f(x) = 2x + 6 \). Make sure the line extends across the graph and through both plotted points.

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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 a way of writing a linear equation. It's written as \( y = mx + b \). This form is incredibly helpful when graphing because it immediately tells us two things: the slope \( m \) and the y-intercept \( b \). The slope \( m \) indicates the steepness or inclination of the line and shows how fast the line rises or falls. The y-intercept \( b \) is the point where the line crosses the y-axis. This form simplifies the process of graphing linear functions as it directly provides key information for plotting.
Slope of a Line
The slope of a line is a measure of its steepness. It is calculated as the ratio of the change in y-values to the change in x-values between any two points on the line. This is often described as "rise over run." For example, a slope \( m = 2 \) means that for each unit the line runs horizontally (rightward), it rises by 2 units. In this example function, \( f(x) = 2x + 6 \), the slope is 2, indicating a constant upwards movement. Knowing the slope is essential for predicting how the line will extend across the graph.
Y-Intercept
The y-intercept is a critical component of a linear equation, specifically in the slope-intercept form. It is the value of y where the line crosses the y-axis, meaning it is the y-coordinate when \( x = 0 \). In the given function, \( f(x) = 2x + 6 \), the y-intercept is 6. This tells us the line will cross the y-axis at the point \( (0, 6) \). Plotting this point is usually the first step in graphing as it provides a starting position for the line.
Plotting Points on a Graph
Plotting points is a simple yet essential skill in graphing linear functions. To plot the function \( f(x) = 2x + 6 \), begin by marking the y-intercept on the graph. Place a point at \( (0, 6) \). Then, use the slope to find another point. Since the slope is 2, move 2 units up and 1 unit to the right from \( (0, 6) \) to find the point \( (1, 8) \). Plot this second point. Connect these points with a straight line, extending it across the graph. This line visually represents the equation and its solutions, providing an easy reference for understanding the function's behavior.

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