Write y = -2x in standard form. Then graph the function.
The Correct Answer and Explanation is:
To convert the equation ( y = -2x ) into standard form, we need to rearrange it into the form ( Ax + By = C ), where ( A ), ( B ), and ( C ) are integers, and ( A ) should be a non-negative integer.
Step 1: Rearranging the Equation
Starting with the equation:
[
y = -2x
]
We can add ( 2x ) to both sides to get:
[
2x + y = 0
]
Step 2: Standard Form
Now, we have the equation in standard form:
[
2x + y = 0
]
Step 3: Graphing the Function
To graph the equation ( y = -2x ) or ( 2x + y = 0 ), we can find two points by substituting values for ( x ) and solving for ( y ).
- Finding Points:
- When ( x = 0 ):
[
y = -2(0) = 0 \quad \Rightarrow \quad (0, 0)
] - When ( x = 1 ):
[
y = -2(1) = -2 \quad \Rightarrow \quad (1, -2)
] - When ( x = -1 ):
[
y = -2(-1) = 2 \quad \Rightarrow \quad (-1, 2)
]
- Plotting the Points:
- Plot the points ( (0, 0) ), ( (1, -2) ), and ( (-1, 2) ) on the Cartesian plane.
- Drawing the Line:
- Connect the points with a straight line. The line will slope downward, reflecting the negative coefficient of ( x ) in the equation.
Explanation of the Graph
The graph of the equation ( y = -2x ) depicts a straight line with a slope of (-2), which indicates that for every unit increase in ( x ), ( y ) decreases by 2 units. The line intersects the y-axis at the origin ((0, 0)), which is the y-intercept. Since the slope is negative, the line will move downwards from left to right, demonstrating the inverse relationship between ( x ) and ( y ) in this linear equation. The standard form ( 2x + y = 0 ) is useful in easily identifying the slope and intercepts of the line, making it applicable for various mathematical analyses and real-world applications.