/*! 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 47 The graph of \(y=5 x\) is given ... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 47

The graph of \(y=5 x\) is given below as well as Figures a-d. Match each equation with its graph. Hint. Recall that if an equation is written in the form \(y=m x+b\) its graph crosses the \(y\) -axis at \((0, b)\) (GRAPH CANNOT COPY) $$ y=5 x+5 $$

Short Answer

Expert verified
Match the graph that intersects at (0, 5) with a steep upward slope.

Step by step solution

01

Identify the parameters

The given equation is in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, \(m = 5\) and \(b = 5\).
02

Interpret the slope

The slope \(m = 5\) means that for every 1 unit increase in \(x\), \(y\) increases by 5 units. This tells us how steep the line is.
03

Determine the y-intercept

The y-intercept \(b = 5\) means that the graph of the equation will cross the y-axis at the point (0, 5).
04

Sketch the graph

Using the slope and y-intercept, start at the point (0, 5) on the y-axis. From there, use the slope to plot another point by moving up 5 units and right 1 unit to find the next point (1, 10).
05

Match the graph

Look at each of the given graphs (a-d). Find the graph that crosses the y-axis at (0, 5) and goes upwards with a slope that moves 5 units up for each 1 unit it moves to the right.

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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 an important concept in algebra that provides a visual representation of mathematical relationships. This involves drawing a line that represents an equation of the form \(y = mx + b\). Understanding how to graph these equations can help you quickly interpret the relationship between the variables.

To effectively graph a linear equation:
  • Identify the equation's slope and y-intercept.
  • Plot the y-intercept on the graph first, as it is the easiest point to identify.
  • Use the slope to find additional points on the graph by moving up or down and side-to-side from the y-intercept.
The line is typically straight, indicating a constant rate of change. By graphing linear equations, you can solve systems of linear equations and find solutions visually.
Slope-intercept form
The slope-intercept form of a linear equation is expressed as \(y = mx + b\). This format makes it convenient to graph a line and understand its characteristics quickly. Here, \(m\) stands for the slope of the line, and \(b\) represents the y-intercept.

The slope \(m\) is a crucial part of this form.
  • The slope indicates the steepness of the line.
  • It shows how much \(y\) changes when \(x\) increases by one unit.
  • A positive slope means the line ascends, while a negative slope shows a descending line.
Meanwhile, the y-intercept \(b\) is where the line crosses the y-axis. This is the value of \(y\) when \(x = 0\). With slope-intercept form, graphing becomes a simple process, making it easier for anyone to understand the relationship between \(x\) and \(y\).
Y-intercept
The y-intercept of a linear equation in the form \(y = mx + b\) is the point where the line crosses the y-axis. This point has coordinates \((0, b)\). The y-intercept provides a starting point for graphing the line since it is the position where the line meets the vertical axis.

In the equation \(y = 5x + 5\), the y-intercept is 5. This means the graph will pass through the point \((0, 5)\), indicating where the line will start on the y-axis.
  • To locate the y-intercept, look at the constant term \(b\) in slope-intercept form.
  • This point is critical in plotting the initial point of your graph.
  • The y-intercept can help you understand the initial value of a relationship, particularly when comparing two lines.
Understanding the y-intercept is fundamental in graphing because it serves as a fixed position from which you can use the slope to find additional points.

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