y=-1/2|x|
The equation \( y = -\frac{1}{2}|x| \) represents a V-shaped graph that opens downwards due to the negative coefficient in front of the absolute value. Here are some key points regarding this equation:
Characteristics:
Vertex: The vertex of the graph is at the origin \((0, 0)\) since there are no horizontal or vertical shifts.
Slope:
- For \( x \geq 0 \), \( |x| = x \) so the equation becomes \( y = -\frac{1}{2}x \). This indicates that the slope of the line is \(-\frac{1}{2}\) in the first quadrant.
- For \( x < 0 \), \( |x| = -x \) so the equation becomes \( y = \frac{1}{2}x \). This indicates that the slope of the line is \( \frac{1}{2}\) in the second quadrant.
Y-Intercept: The y-intercept occurs when \( x = 0 \). Here, \( y = 0 \), so the point is \((0, 0)\).
X-Intercepts: The only x-intercept occurs at \( y=0 \). Setting \( y = 0 \) gives \( |x| = 0 \), which means \( x = 0\).
Symmetry: The graph is symmetric about the y-axis because it contains the absolute value function which is even.
Graph:
To sketch the graph:
- Start by plotting the vertex at the origin (0, 0).
- From the vertex, draw lines with slopes of \(-\frac{1}{2}\) to the right and \(\frac{1}{2}\) to the left.
- Extend the lines indefinitely to show that they continue in both directions.
Summary:
The graph of \( y = -\frac{1}{2}|x| \) is a V-shape, opening downwards with its vertex at the origin and symmetric about the y-axis. The lines have slopes of \(-\frac{1}{2}\) and \(\frac{1}{2}\) for \( x \geq 0\) and \( x < 0\) respectively.
