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Chapter 4: Problem 61

In the following exercises, graph by plotting points. \(y=-2 x+2\)

Short Answer

Expert verified
Plot points (0, 2), (1, 0), and (-1, 4) and draw a line through them.

Step by step solution

01

- Choose some values for x

Pick at least two values for x to find corresponding y values. Commonly used values are 0, 1, and -1.
02

- Calculate the corresponding y values

Substitute the chosen x values into the equation to find the corresponding y values. For example, if x = 0: y = -2(0) + 2 = 2If x = 1: y = -2(1) + 2 = 0If x = -1: y = -2(-1) + 2 = 4
03

- Plot the points

Plot the points (0, 2), (1, 0), and (-1, 4) on a coordinate plane.
04

- Draw the line

Connect the plotted points with a straight line. This line represents the equation y = -2x + 2.

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Key Concepts

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

plotting points
Plotting points is a fundamental skill in graphing linear equations. It involves identifying specific coordinates on a graph to represent values from an equation. Each point has an 'x' (horizontal) and 'y' (vertical) value. To plot a point, you find the given x-value on the horizontal axis and then trace it vertically to meet the corresponding y-value.

For example, in the exercise solution, the points (0, 2), (1, 0), and (-1, 4) were plotted. This means:
  • (0, 2) is at x=0 and y=2
  • (1, 0) is at x=1 and y=0
  • (-1, 4) is at x=-1 and y=4

Follow these steps:
  • Start at the origin (0,0) on the graph.
  • Move horizontally to the x-value.
  • From the x-value, move vertically to the y-value.
  • Mark the spot where you stopped.
By plotting these points, you are laying the groundwork for drawing the linear equation on the coordinate plane.
coordinate plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface used in graphing linear equations. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis), which intersect at a point called the origin (0,0).

Here are key components of a coordinate plane:
  • X-axis: Runs left to right (horizontal) and represents the independent variable.
  • Y-axis: Runs top to bottom (vertical) and represents the dependent variable.
  • Quadrants: The plane is divided into four quadrants. Quadrant I (top-right), Quadrant II (top-left), Quadrant III (bottom-left), and Quadrant IV (bottom-right).

Plotting points on this plane allows us to visualize the equation y = -2x + 2. Each point is defined by a pair of values (x, y). By marking these points and connecting them, you create a straight line that represents the equation. Understanding the coordinate plane is crucial for graphing any kind of function or equation.
linear equations
A linear equation is an equation that forms a straight line when graphed. It can be written in various forms, with the slope-intercept form being the most common, represented as y = mx + b, where 'm' is the slope and 'b' is the y-intercept.

Here, the equation y = -2x + 2 follows this form:
  • Slope (m): The slope is -2. This means for every unit increase in x, y decreases by 2 units.
  • Y-intercept (b): The y-intercept is 2. This is the point where the line crosses the y-axis when x=0.

Steps to graph a linear equation:
  • Choose values for x (e.g., 0, 1, -1).
  • Calculate corresponding y values using the equation.
  • Plot the resulting points on the coordinate plane.
  • Connect the dots with a straight line.

Linear equations like y = -2x + 2 are simple to graph once you understand these principles. They are fundamental in algebra and essential for advancing in math.

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Most popular questions from this chapter

Find the equation of each line. Write the equation in slope-intercept form. Perpendicular to the line \(y-1=0\), point (-2,6)

Determine whether each ordered pair is a solution to the inequality \(x+y>4\) : (a) (5,1) (b) (-2,6) c) (3,2) (d) (10,-5) (e) (0,0)

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(y=5,\) point (2,-2)

Graph the linear inequality \(y>\frac{2}{3} x-1\)

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(x=-4,\) point (-3,-5)
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