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Chapter 1: Problem 50

Use the y-intercept and slope to sketch the graph of each equation. $$y=-x+3$$

Short Answer

Expert verified
Plot the y-intercept (0, 3) and use the slope -1 to find and plot (1, 2). Draw the line through these points.

Step by step solution

01

Identify the y-intercept

The y-intercept of the equation can be found by looking at the constant term. In the equation \(y = -x + 3\), the y-intercept is 3. This means the graph will cross the y-axis at the point (0, 3).
02

Determine the slope

The slope of a line in the form \(y = mx + b\) is represented by the coefficient of x. In the equation \(y = -x + 3\), the slope (m) is -1. This means the line decreases by 1 unit on the y-axis for every 1 unit it moves to the right on the x-axis.
03

Plot the y-intercept

Plot the point (0, 3) on the coordinate plane. This is where the line will intersect the y-axis.
04

Use the slope to find another point

Starting from the y-intercept (0, 3), use the slope to find another point on the line. With a slope of -1, move 1 unit to the right (positive direction on the x-axis) and 1 unit down (negative direction on the y-axis). This gives the point (1, 2). Plot this point.
05

Draw the line

Draw a straight line through the points (0, 3) and (1, 2). This is the graph of the equation \(y = -x + 3\).

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Key Concepts

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

y-intercept
In a linear equation, the y-intercept is where the line crosses the y-axis. This is the value of y when x is zero. In the equation \(y = -x + 3\), the y-intercept is 3. This means the graph will intersect the y-axis at the point (0, 3). Identifying the y-intercept is crucial for graphing because it gives a fixed starting point on the coordinate plane.
slope
The slope of a line indicates its steepness and direction. It is the ratio of the change in y to the change in x. In the equation \(y = -x + 3\), the slope is -1. This means for each unit increase in x, y decreases by 1 unit.
To visualize this:
  • Start at the y-intercept (0, 3).
  • Move 1 unit to the right along the x-axis (increasing x by 1).
  • Move 1 unit down along the y-axis (decreasing y by 1).
This gives us another point on the line (1, 2). The slope makes it easy to find additional points for drawing the line.
coordinate plane
The coordinate plane is a two-dimensional surface on which we plot points, lines, and curves. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
Each point on the plane is represented by a pair of coordinates \((x, y)\). For instance, the y-intercept (0, 3) and another point determined by the slope (1, 2) can be plotted on this plane.
Visualizing these points on the coordinate plane helps us to draw the entire line by simply connecting them.
equation of a line
The equation of a line in slope-intercept form is \(y = mx + b\). Here:
  • m represents the slope.
  • b represents the y-intercept.
For the equation \(y = -x + 3\), the slope (m) is -1, and the y-intercept (b) is 3. Knowing these two elements enables us to graph the line accurately.
Simply start at the y-intercept, then use the slope to find another point. Finally, draw a straight line through these points to represent the equation visually.

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