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Magazine Chapter 2: Problem 3
Sketch the graph of the equation, and label the \(x\) - and \(y\) -intercepts. $$y=-x+2$$
Short Answer
Expert verified
The x-intercept is (2, 0) and the y-intercept is (0, 2); plot these points and connect them with a line.
Step by step solution
01
Identify the Type of Equation
The equation is in the form of a linear equation, which can be represented as \(y = mx + b\). Here, \(m\) is the slope and \(b\) is the y-intercept. In the equation \(y = -x + 2\), the slope \(m = -1\) and the y-intercept \(b = 2\).
02
Find the Y-Intercept
The y-intercept is the point where the graph crosses the y-axis. Plugging in \(x = 0\) into the equation \(y = -x + 2\), we find that \(y = 2\). Therefore, the y-intercept is at the point \((0, 2)\).
03
Find the X-Intercept
The x-intercept is the point where the graph crosses the x-axis. This is obtained by setting \(y = 0\) in the equation \(y = -x + 2\). Solving \(0 = -x + 2\) gives \(x = 2\). Therefore, the x-intercept is at the point \((2, 0)\).
04
Plot the Intercepts
On a graph, plot the y-intercept at \((0, 2)\) and the x-intercept at \((2, 0)\). These points will help in drawing the line of the graph.
05
Draw the Line
Using a ruler, connect the x-intercept and y-intercept with a straight line. This line represents the graph of the equation \(y = -x + 2\).
06
Label the Intercepts
Clearly label the x-intercept at \((2, 0)\) and the y-intercept at \((0, 2)\) on the graph. This ensures the intercepts are easy to identify.
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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 involves plotting points on a coordinate plane that describe the solutions to the equation. The equation, often in the form of \( y = mx + b \), is a straight line on the graph. Each point on this line is a solution where the coordinates \((x, y)\) satisfy the equation.
To draw the graph of a linear equation:
To draw the graph of a linear equation:
- Identify key points: like the x-intercept and y-intercept.
- Plot these points carefully on the coordinate plane.
- Use a ruler to draw a straight line through these points, extending it across the graph.
x-intercept
The x-intercept is the point where a graph crosses the x-axis. At this point, the value of \( y \) is zero because it is where the line meets the horizontal axis. To find the x-intercept, you simply set \( y = 0 \) in the equation and solve for \( x \).
For example, in the equation \( y = -x + 2 \), you set \( y = 0 \):
This point provides a crucial anchor for drawing the graph.
For example, in the equation \( y = -x + 2 \), you set \( y = 0 \):
- 0 = -x + 2
- Add \( x \) to both sides: \( x = 2 \)
This point provides a crucial anchor for drawing the graph.
y-intercept
The y-intercept is the point where a graph crosses the y-axis. Here, \( x \) is zero since it is where the line meets the vertical axis. You can find the y-intercept by setting \( x = 0 \) in the equation and solving for \( y \).
Consider the equation \( y = -x + 2 \), set \( x = 0 \) and find:
Like the x-intercept, this point is used to help draw the line of the graph.
Consider the equation \( y = -x + 2 \), set \( x = 0 \) and find:
- \( y = -0 + 2 \)
- Simplify to \( y = 2 \)
Like the x-intercept, this point is used to help draw the line of the graph.
slope-intercept form
The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This format is particularly useful because it quickly tells you the slope of the line and where it crosses the y-axis.
The slope \( m \) represents the rate of change, indicating how steep the line is:
The slope \( m \) represents the rate of change, indicating how steep the line is:
- Positive slope means the line rises as it moves from left to right.
- Negative slope means the line falls as it moves from left to right.
coordinate plane
The coordinate plane is a two-dimensional surface used for graphing equations. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis, intersecting at a point called the origin (0,0).
Each point on the plane is defined by an ordered pair \((x, y)\), representing its position relative to the axes. For example:
Each point on the plane is defined by an ordered pair \((x, y)\), representing its position relative to the axes. For example:
- The point \((2, 0)\) lies on the x-axis two units to the right of the origin.
- The point \((0, 2)\) is on the y-axis, two units above the origin.
