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Chapter 0: Problem 51

Sketch a graph of the equation. $$y=-2 x+1$$

Short Answer

Expert verified
The graph of the equation \(y=-2x+1\) is a straight line on a graph passing through the points (0,1) and (1,-1). The line slopes downward from left to right, illustrating a slope of -2.

Step by step solution

01

Identify the y-intercept and plot it

The y-intercept is the point where the line crosses the y-axis. This happens when \(x=0\). In the equation \(y=-2x+1\), with \(x=0\), we get \(y=1\). Hence, the y-intercept is at point (0,1). So, the first point to be plotted on the graph is (0,1).
02

Identify the slope and plot another point

The slope of the given line is -2, which is the coefficient of \(x\). The slope is typically represented as a ratio of 'rise over run', meaning for every 2 units down (a negative rise), you move 1 unit to the right. From the y-intercept (0,1), move 2 units down and 1 unit to the right to find a second point (1,-1). This point should also be plotted on the graph.
03

Draw the line

Once you have at least two points, you can plot the line of the equation. Draw a straight line that passes through these two points. This line represents the set of solutions for the equation \(y=-2x+1\).

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Key Concepts

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

Understanding the Y-Intercept
The y-intercept is an essential component of a linear equation and is found where the graph crosses the y-axis. To find this point, set the value of x to zero in the equation. In the given equation \(y = -2x + 1\), this results in:
  • \(x = 0\)
  • \(y = 1\)
Thus, the y-intercept is at the coordinate point (0, 1). This means that no matter the slope, the line will always start from this point on the graph when x equals zero. As a starting point for your graph, the y-intercept is crucial and makes plotting linear equations simpler.
Deciphering the Slope
The slope of a line indicates its steepness and direction. It is the coefficient of \(x\) in the equation. In this instance, it is \(-2\). This slope tells us:
  • For every unit of movement to the right on the x-axis, the line goes down by 2 units on the y-axis.
  • The negative sign shows a downward slope from left to right.
The slope can be described in terms of 'rise' over 'run'. Here, the 'rise' is -2 and the 'run' is 1. Understanding the slope is key to plotting more points on the graph beyond the y-intercept.
Importance of Coordinate Points
Coordinate points are pairs of values that give the exact position of points on a graph. Each point is defined by an \((x, y)\) pair.When graphing, begin with known points like the y-intercept. For the equation \(y = -2x + 1\), start at the coordinate point (0, 1).Next, apply the slope to find additional points. From (0, 1), move down 2 units and right 1 unit to find the point (1, -1).Finding multiple points ensures accuracy in your graph. You can continue using the slope to find more points, such as moving from (1, -1) to (2, -3), ensuring they align in a straight line.
Guide to Plotting Graphs
Plotting graphs becomes straightforward once you understand your equation's y-intercept and slope. Follow these steps:
  • Start by plotting the y-intercept, here at (0, 1).
  • Use the slope to determine additional points. Move as the slope indicates (e.g., down 2, right 1 from (0, 1) to (1, -1)).
  • Double-check your points to ensure a straight line can pass through them all.
Connect these points with a ruler or straight edge to draw a line. This line visually represents all the solutions of the equation. Your graph should neatly show all calculated points along the line, confirming accuracy and understanding.

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Most popular questions from this chapter

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