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Chapter 2: Problem 2

Exer. \(1-20\) : Sketch the graph of the equation, and label the \(x\) - and \(y\) -intercepts. $$y=3 x+2$$

Short Answer

Expert verified
Graph is a straight line with x-intercept at \((-\frac{2}{3}, 0)\) and y-intercept at \((0, 2)\).

Step by step solution

01

Identify the Equation Type

The given equation is in the form \(y = mx + b\), which is a linear equation. Here, \(m = 3\) is the slope, and \(b = 2\) is the y-intercept.
02

Calculate X-Intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). \(0 = 3x + 2\). Subtract 2 from both sides to get \(-2 = 3x\). Divide both sides by 3 to find \(x = -\frac{2}{3}\). Therefore, the x-intercept is \((-\frac{2}{3}, 0)\).
03

Identify Y-Intercept

The y-intercept is the constant term in the linear equation, \(b = 2\). Thus, the y-intercept of the graph is \((0, 2)\).
04

Plot the Intercepts

On graph paper, mark the x-intercept \((-\frac{2}{3}, 0)\) and the y-intercept \((0, 2)\). These points will be used to draw the graph.
05

Draw the Line through the Intercepts

Using a ruler, draw a straight line that passes through the points \((-\frac{2}{3}, 0)\) and \((0, 2)\). This line represents the graph of the equation \(y = 3x + 2\).
06

Label the Intercepts

Clearly label the x-intercept and y-intercept on the graph. Write \((-\frac{2}{3}, 0)\) near the point where the graph crosses the x-axis and \((0, 2)\) where it crosses the y-axis.

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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 helps us visualize solutions to algebraic expressions. A linear equation in two variables, such as the form \(y = mx + b\), represents a straight line when graphed on a coordinate plane. The key components involved in graphing these equations are the slope \(m\) and the y-intercept \(b\). To accurately draw the graph of a linear equation, one needs to identify points through which the line passes and then connect these points with a straight edge.
  • **Choose two points**: Typically, intercepts (where the line crosses the axes) are chosen for simplicity.
  • **Mark on the graph**: Place the chosen intercept points accurately on the coordinate plane.
  • **Connect the dots**: Use a ruler to draw a line through these points to ensure accuracy.
Graphing is a fundamental skill in algebra, offering a visual representation of equations that can make understanding the behavior of the line more intuitive.
slope-intercept form
The slope-intercept form is a simple way to express linear equations, represented by the formula \(y = mx + b\). In this equation:
  • \(m\) represents the slope of the line, indicating its steepness and direction.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.
This form is particularly advantageous because it immediately provides the slope and y-intercept, allowing you to quickly plot the graph of the line.
The slope \(m\) shows how the y-coordinate changes for a given change in the x-coordinate, indicating how steep the line is. A positive slope means an upward incline from left to right, while a negative slope indicates a downward incline.
  • **Positive slope**: As x increases, y increases.
  • **Negative slope**: As x increases, y decreases.
x-intercept
The x-intercept is a critical point on a graph where the line crosses the x-axis. At this position, the y-coordinate is zero. To find the x-intercept in the equation, \(y = 3x + 2\), you set \(y\) to zero and solve for \(x\). In our specific exercise:
  • Substitute \(y = 0\): \(0 = 3x + 2\)
  • Solve for \(x\): \(3x = -2\) leads to \(x = -\frac{2}{3}\)
Thus, the x-intercept is \((-\frac{2}{3}, 0)\). This point provides a crucial piece of information for sketching the graph, as it is one of the two key points that define the line.
y-intercept
The y-intercept is where the line crosses the y-axis on a graph, represented by the point where \(x = 0\). In the slope-intercept form \(y = mx + b\), the y-intercept is given by \(b\). For our equation \(y = 3x + 2\), this means:
  • When \(x = 0\), \(y = 2\)
  • Therefore, the y-intercept is \((0, 2)\)
This point is crucial as it provides an easily identifiable starting place when graphing. It dictates where the line will begin on the y-axis, enabling you to plot the line effectively on a graph.

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