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Chapter 2: Problem 28

Graph each equation. \(y=2 x-3\)

Short Answer

Expert verified
Plot the point (0, -3), use slope 2 to plot (1, -1), and draw the line through these points.

Step by step solution

01

Identify Equation Type

The given equation is in the form of a linear equation: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For this equation, \(m = 2\) and \(b = -3\).
02

Plot the Y-Intercept

The y-intercept \(b\) is -3. This means that the graph will cross the y-axis at \( (0, -3) \). Plot this point on the graph.
03

Use the Slope to Find Another Point

The slope \(m\) is 2, which means that for every horizontal increase of 1 unit, the vertical increase is 2 units. Starting from \((0, -3)\), move right 1 unit to \(x = 1\), then up 2 units to \(y = -1\). Plot this point \((1, -1)\) on the graph.
04

Draw the Line

Using a ruler, draw a straight line through the points \((0, -3)\) and \((1, -1)\). Extend this line in both directions, ensuring it has a consistent slope of 2.

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Key Concepts

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

Graphing Equations
Linear equations, such as \(y = mx + b\), can be visualized by plotting their graphs on a coordinate plane. This is an effective way to understand the relationship between the variables \(x\) and \(y\). To graph a linear equation:
  • Start by identifying the y-intercept \(b\). This is where the line will cross the y-axis.
  • Next, use the slope \(m\) to determine the direction and steepness of the line.
  • Lastly, plot at least two points and draw a straight line through them to complete the graph.
Graphing equations provides a visual understanding of how changes in \(x\) affect \(y\), which is a foundational concept in algebra. By representing linear equations graphically, students can intuitively grasp how these basic functions behave.
Slope-Intercept Form
The slope-intercept form of a linear equation is expressed as \(y = mx + b\). Understanding each component is key:
  • The slope \(m\) indicates how steep the line is. It's the rate of change of \(y\) with respect to \(x\).
  • The y-intercept \(b\) gives us a starting point on the y-axis.
To remember, \(m\) determines the tilt of the line:
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope makes the line fall as it moves left to right.
  • A slope of zero results in a horizontal line.
This form makes it straightforward to graph a line because it breaks down the equation into easily identifiable starting points and directions. Recognizing \(m\) and \(b\) in equations quickly allows students to plot lines efficiently.
Y-Intercept
The y-intercept of an equation is the point where the line crosses the y-axis. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\). This point is crucial because:
  • It gives a fixed point to start graphing.
  • It's the value of \(y\) when \(x\) is zero.
For example, if \(b = -3\), the line crosses the y-axis at \(y = -3\), or \((0, -3)\). By knowing the y-intercept, you can immediately plot one point on the graph without needing to calculate anything further. This simplifies constructing the rest of the graph and helps you quickly visualize the line's position in the coordinate plane. So, next time when you come across an equation, look for \(b\) to efficiently start your graph.

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Most popular questions from this chapter

Fill in the blanks. a. \(m=\frac{\text { }}{\text { horizontal change }}=\frac{\text { change in } y}{\Delta x}=\frac{ }{\Delta x}\) b. \(m=\frac{y_{2}-y_{1}}{ }\)

Find the slope and -intercept and use them to draw the graph of the line. $$ y=-x+2 $$

Graph each equation \(y=1.5 x-4\)

Write an equation for a linear function whose graph has the given characteristics. Passes through ( \(3,0\) ), parallel to the graph of \(g(x)=-\frac{2}{3} x-4\)

Write each equation in slope鈥搃ntercept form. Then find the slope and the y-intercept of the line determined by the equation. $$ 3 x-2 y=8 $$
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