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Chapter 7: Problem 4

Graph each equation using the slope and \(y\) -intercept. $$y=\frac{1}{4} x+1$$

Short Answer

Expert verified
Plot the y-intercept at \((0, 1)\) and use the slope \(\frac{1}{4}\) to find another point, then draw the line through these points.

Step by step solution

01

Identify the Slope and Y-intercept

The given equation is in slope-intercept form, which is \(y = mx + b\). Here, \(m\) is the slope and \(b\) is the y-intercept. From the equation \(y=\frac{1}{4}x+1\), the slope \(m\) is \(\frac{1}{4}\) and the y-intercept \(b\) is \(1\).
02

Plot the Y-intercept

Start by plotting the y-intercept of the equation on the graph. Since the y-intercept \(b\) is \(1\), place a point at \((0, 1)\) on the y-axis.
03

Use the Slope to Find Another Point

The slope \(\frac{1}{4}\) means for every 1 unit increase in \(x\), \(y\) increases by \(\frac{1}{4}\) unit. From the y-intercept \((0,1)\), move 1 unit to the right to \((1,1)\) and then move up \(\frac{1}{4}\) unit to reach \((1,1.25)\). Plot the second point at \((1,1.25)\).
04

Draw the Line Through the Points

Connect the two points you have plotted: \((0, 1)\) and \((1, 1.25)\) with a straight line. Extend the line on both sides beyond these points.
05

Label the Graph

Label the x-axis and y-axis appropriately. Mark the equation \(y=\frac{1}{4}x+1\) on the line to indicate that it represents that specific linear equation.

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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 of a linear equation is one of the easiest ways to represent and understand a line. It is written as: \[ y = mx + b \]Here:
  • \( y \) represents the dependent variable or the output of the function.
  • \( m \) is the slope of the line, indicating its steepness and direction.
  • \( x \) is the independent variable or input of the function.
  • \( b \) is the y-intercept, which is the value at which the line crosses the y-axis.
This form is very useful because it immediately shows you the slope and y-intercept. You can quickly graph the equation by using these two values, making it easy to visualize how the line behaves.
Plotting Points
Plotting points on a graph is a way to represent data in a visual format. For a linear equation, you start by locating key points that satisfy the equation. To plot points for the line represented by the equation \( y = \frac{1}{4}x + 1 \):
  • Begin with the y-intercept, which is \( (0, 1) \). This is the starting point on the y-axis.
  • Use the slope to determine additional points. For every 1 unit you move to the right along the x-axis, you'll move up \( \frac{1}{4} \) on the y-axis. This process finds the next point by moving in line with the slope.
  • For example, from \( (0, 1) \) move to \( (1, 1.25) \) by applying the slope \( \frac{1}{4} \).
Using these methods, you can accurately draw your line.
Finding the Slope
The slope of a line tells you how steep the line is and in which direction it is going. Mathematically, the slope \( m \) is the ratio of the change in the y-values to the change in the x-values, often expressed as 'rise over run.' For example, if a line has a slope \( \frac{1}{4} \), it means:
  • For every increase of 1 unit in the x-direction, the y-value increases by \( \frac{1}{4} \).
  • A positive slope means the line rises to the right, while a negative slope means it falls to the right.
In our equation \( y = \frac{1}{4}x + 1 \), the slope is \( \frac{1}{4} \), indicating a gentle incline as the line moves to the right.
Y-Intercept
The y-intercept of a line is a crucial point for graphing, as it shows where the line crosses the y-axis. This point is found directly from the equation in slope-intercept form \( y = mx + b \). The y-intercept \( b \) is the constant in the equation.
  • For \( y = \frac{1}{4}x + 1 \), the y-intercept is \( 1 \).
  • This means the line crosses the y-axis at \( (0, 1) \).
Understanding the y-intercept helps in quickly setting up a graph, as it provides a fixed reference point. From there, you can use the slope to draw the line accurately.

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Most popular questions from this chapter

Graph each equation using the slope and \(y\) -intercept. $$y=-3$$

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Evaluate each expression if \(x=4\) and \(y=-1.\) $$2 y$$
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