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Magazine Chapter 2: Problem 43
Find the \(x\) - and \(y\) -intercepts and use them to graph the following functions. $$-x+12 y=6$$
Short Answer
Expert verified
Intercepts: x = -6, y = 0.5. Graph using intercepts.
Step by step solution
01
Find the x-intercept
To find the x-intercept, set \(y = 0\) in the equation. This gives us:\[-x + 12(0) = 6\]Simplifying, we have:\[-x = 6\]So, dividing both sides by -1, we find the x-intercept is:\[x = -6\]The x-intercept is \((-6, 0)\).
02
Find the y-intercept
To find the y-intercept, set \(x = 0\) in the equation. This gives us:\[-(0) + 12y = 6\]Simplifying, we have:\[12y = 6\]Dividing both sides by 12, we find:\[y = \frac{1}{2}\]So, the y-intercept is \((0, \frac{1}{2})\).
03
Plot the intercepts
Use the intercepts \((-6, 0)\) and \((0, \frac{1}{2})\) to plot the line on the graph. Plot the point \((-6, 0)\) on the x-axis, and the point \((0, \frac{1}{2})\) on the y-axis.
04
Draw the line
Connect the two intercept points with a straight line. This line represents the graph of the equation \(-x + 12y = 6\). Extend the straight line across the graph to represent the function over its entire domain.
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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 of a linear equation is the point where the graph of the equation crosses the x-axis. At this point, the y-coordinate is always 0. To find the x-intercept, simply set the y-value to 0 in the equation and solve for x. This makes sense because on the x-axis, the height (or y-value) of any point is zero.In our example equation \[-x + 12y = 6\], when we set \(y = 0\), we solve \[-x + 12(0) = 6\] to \[-x = 6\]. By dividing each side by -1, we find \(x = -6\). Thus, the x-intercept is the point \((-6, 0)\).Key Points:
- The x-intercept is found by setting y to 0 in the equation.
- It is a crucial part of graphing linear equations, helping to anchor one point of the line on the graph.
- This intercept represents where the line touches the x-axis.
Understanding the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. At this point, the x-coordinate is always 0. This is because the y-axis is the vertical axis where x is zero.To find the y-intercept of an equation, set x to 0 and solve for y. Let's look at the equation from the exercise,\[-x + 12y = 6\]. Setting \(x = 0\), we have:\[-(0) + 12y = 6\] yielding \[12y = 6\]. Dividing both sides by 12 gives \(y = \frac{1}{2}\).Thus, the y-intercept is the point \((0, \frac{1}{2})\).Significance of the Y-Intercept:
- It indicates where the line touches the y-axis.
- It gives another key point to plot when graphing linear equations.
- It's critical for understanding the general position of the line.
Graphing Linear Equations
Graphing linear equations involves drawing a line on the coordinate plane that represents all the solutions of the equation. Linear equations will always graph to form a straight line, hence the name "linear."Using intercepts simplifies this process:
- Find the x- and y-intercepts, as explained in the earlier sections.
- Plot these points on the coordinate plane鈥攐ne on the x-axis and one on the y-axis.
- Connect the intercepts with a straight line extending across the graph.
- Graphing helps visualize the relationship between variables in the equation.
- It demonstrates how changes in one variable affect the other.
- Graphs make it easier to predict future values and understand trends.
The Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is a flat surface defined by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Where they intersect is known as the origin, \((0, 0)\).Understanding and using the coordinate plane is essential for graphing linear equations:
- Each point on the plane is defined by a pair \((x, y)\).
- It helps to locate and plot points, such as intercepts, quickly.
- The grid allows for the visual representation of equations as lines.
- It allows for a better understanding of spatial relationships between points.
- The plane helps in analyzing and interpreting the graph of an equation.
- It acts as a reference system to solve and graph equations accurately.
