Chapter 1: Problem 29
If the equation of a line is \(y=\frac{1}{2} x+3\), then mark on the graph the point where the line crosses the \(y\)-axis and the point where the line crosses the \(x\)-axis.
Short Answer
Expert verified
The line crosses the y-axis at the point (0, 3) and crosses the x-axis at the point (-6, 0).
Step by step solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the X-Intercept
The x-intercept is a critical point where a line crosses the x-axis on a graph. For any linear equation like \[ y = mx + b \], the x-intercept can be found by setting \( y = 0 \) and solving for \( x \).
In the equation \( y = \frac{1}{2}x + 3 \), we find the x-intercept by substituting \( y = 0 \): \[0 = \frac{1}{2}x + 3\]
Solving this gives: \[-3 = \frac{1}{2}x\] Multiplying both sides by 2: \[-6 = x\].
The x-intercept is the point \((-6, 0)\) on the graph. This indicates where the line cuts through the x-axis. Identifying the x-intercept helps in understanding the behavior of the line as it provides key information about its position and slope.
In the equation \( y = \frac{1}{2}x + 3 \), we find the x-intercept by substituting \( y = 0 \): \[0 = \frac{1}{2}x + 3\]
Solving this gives: \[-3 = \frac{1}{2}x\] Multiplying both sides by 2: \[-6 = x\].
The x-intercept is the point \((-6, 0)\) on the graph. This indicates where the line cuts through the x-axis. Identifying the x-intercept helps in understanding the behavior of the line as it provides key information about its position and slope.
- Set \( y = 0 \)
- Solve for \( x \)
- Mark point \((-6, 0)\)
Understanding the Y-Intercept
The y-intercept is the point on a graph where the line crosses the y-axis. It tells you where the line starts if you were to draw it from the intercept point. In the equation \( y = mx + b \), the parameter \( b \) represents the y-intercept. To find it, set \( x = 0 \) and solve for \( y \).
In our example, using \( y = \frac{1}{2}x + 3 \), substitute \( x = 0 \):\[y = \frac{1}{2}(0) + 3\]
This simplifies to \( y = 3 \), which means the y-intercept is at point \((0, 3)\).
This point gives you a place to start drawing the line, and it is essential for defining the line's slope and direction. Whenever tackling a new line equation, finding the y-intercept is always a good initial step.
In our example, using \( y = \frac{1}{2}x + 3 \), substitute \( x = 0 \):\[y = \frac{1}{2}(0) + 3\]
This simplifies to \( y = 3 \), which means the y-intercept is at point \((0, 3)\).
This point gives you a place to start drawing the line, and it is essential for defining the line's slope and direction. Whenever tackling a new line equation, finding the y-intercept is always a good initial step.
- Set \( x = 0 \)
- Calculate \( y \)
- Mark point \((0, 3)\)
Navigating the Cartesian Plane
The Cartesian plane is a two-dimensional graph consisting of the horizontal x-axis and the vertical y-axis. It is used to plot points, lines, and curves, allowing us to visualize mathematical relationships. By understanding how to work with this plane, interpreting equations becomes more intuitive.
Here's a helpful breakdown:
Here's a helpful breakdown:
- Axes: The horizontal line is called the x-axis, and the vertical line is the y-axis. They intersect at the origin \((0, 0)\).
- Quadrants: The plane is divided into four quadrants. Points in Quadrant I have positive x and y coordinates, Quadrant II has negative x and positive y, Quadrant III has both negative, and Quadrant IV has positive x and negative y.
- Plotting Points: Each point on the plane is identified by coordinates \((x, y)\). Moving right or left follows the x-axis, and moving up or down follows the y-axis.
- Lines: Linear equations can be graphed with points found using x- and y-intercepts, helping to visualize their slope and intersection on the Cartesian plane.
