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Chapter 2: Problem 4

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

Short Answer

Expert verified
Graph the equation by joining the y-intercept (0, -3) and the x-intercept \( \left( -\frac{3}{2}, 0 \right) \).

Step by step solution

01

Identify the equation type

The given equation is in the form of a linear equation, which is generally expressed as \( y = mx + c \), where \( m \) represents the slope and \( c \) is the y-intercept.
02

Determine the y-intercept

The y-intercept is the point where the line crosses the y-axis. Set \( x = 0 \) in the equation \( y = -2x - 3 \). This gives \( y = -3 \). Thus, the y-intercept is (0, -3).
03

Find the x-intercept

The x-intercept is the point where the line crosses the x-axis. Set \( y = 0 \) in the equation \( y = -2x - 3 \). Solving \( 0 = -2x - 3 \) yields \( x = -\frac{3}{2} \). Therefore, the x-intercept is \( \left( -\frac{3}{2}, 0 \right) \).
04

Plot the intercepts on a coordinate plane

Place the points (0, -3) and \( \left( -\frac{3}{2}, 0 \right) \) on a coordinate grid. These two points indicate where the line will cross the axes.
05

Sketch the line

Draw a straight line through the points (0, -3) and \( \left( -\frac{3}{2}, 0 \right) \). This line represents the graph of the equation \( y = -2x - 3 \).

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Key Concepts

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

Graphing Linear Functions
Graphing linear functions involves creating a visual representation of a given linear equation on a coordinate plane. A linear function is typically expressed in the form of \( y = mx + c \). In this equation, \( m \) is the slope, and \( c \) is the y-intercept. The graph of a linear function is a straight line.
To sketch a graph of a linear equation, follow these steps:
  • Identify the slope \( m \) and y-intercept \( c \).
  • Determine the intercepts (where the line crosses the axes).
  • Plot these intercepts on a coordinate grid.
  • Draw a straight line through the plotted points.

By plotting the x- and y-intercepts and connecting them with a straight line, you can easily visualize the behavior of the linear equation on the graph.
X-Intercepts
The x-intercept of a linear function is the point where the graph of the equation crosses the x-axis. To find the x-intercept, set \( y = 0 \) in the equation and solve for \( x \). This involves isolating \( x \) to identify the point where the line touches the x-axis.
In the given equation \( y = -2x - 3 \), setting \( y = 0 \) gives us \( 0 = -2x - 3 \). Solving this, we find \( x = -\frac{3}{2} \), thus the x-intercept is \( \left( -\frac{3}{2}, 0 \right) \). The x-intercept is crucial in forming the graph as it provides a precise point where the line interacts with the x-axis. Knowing this helps in accurately plotting and sketching the equation.
Y-Intercepts
The y-intercept of a linear function is the point where the graph crosses the y-axis. This can be easily determined by setting \( x = 0 \) in the equation, which gives the value of \( y \) at that point.
For the equation \( y = -2x - 3 \), substituting \( x = 0 \) directly results in \( y = -3 \). Thus, the y-intercept for this function is \( (0, -3) \). This point is an anchor for graphing as it signifies where the line begins its journey from the y-axis across the graph.
  • Identifying the y-intercept simplifies the plotting process.
  • It provides a reliable starting point for the graph.
Slope
The slope of a linear function indicates the steepness and direction of the line on the graph. It is represented as \( m \) in the equation \( y = mx + c \), denoting the ratio of the vertical change (rise) to the horizontal change (run) \( \frac{\Delta y}{\Delta x} \).
In our equation \( y = -2x - 3 \), the slope \( m \) is \(-2\). This means for every unit increase in \( x \), \( y \) decreases by 2 units, indicating a downward slope from left to right.
The concept of the slope is essential for:
  • Determining how the line tilts (upwards or downwards).
  • Calculating the rate of change between points on the line.
Understanding how the slope works enables you to predict the behavior of the line on the graph effectively.

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Most popular questions from this chapter

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