Question
y = |3x| - 2 graph shape
Answers
GPT-4o mini
The equation \( y = |3x| - 2 \) defines a V-shaped graph, which is typical for absolute value functions. To understand the shape and characteristics of the graph, let's break it down:
1. **Absolute Value Function**: The term \( |3x| \) represents an absolute value function, which creates a V shape. The expression \( |3x| \) is equal to \( 3x \) when \( x \) is positive and \( -3x \) when \( x \) is negative.
2. **Vertical Shift**: The entire graph is shifted downward by 2 units due to the \(-2\) at the end of the equation. This means that the vertex (the point where the V opens) will be located at the point where \( |3x| = 0\), which occurs when \( x = 0 \). Thus, the vertex will be at the point \( (0, -2) \).
3. **Slope**: The slopes of the lines forming the V shape are determined by the coefficient of \( x \) in the absolute value function:
- For \( x \geq 0 \) (the right side of the V), the equation simplifies to \( y = 3x - 2 \).
- For \( x < 0 \) (the left side of the V), the equation simplifies to \( y = -3x - 2 \).
4. **Intercepts**:
- **Y-Intercept**: Substitute \( x = 0 \) into the equation:
\[
y = |3(0)| - 2 = 0 - 2 = -2.
\]
So the y-intercept is at \( (0, -2) \).
- **X-Intercepts**: Set \( y = 0 \):
\[
0 = |3x| - 2 \\
|3x| = 2.
\]
This leads to two equations:
\[
3x = 2 \implies x = \frac{2}{3}
\]
and
\[
3x = -2 \implies x = -\frac{2}{3}.
\]
So the x-intercepts are at \( \left(\frac{2}{3}, 0\right) \) and \( \left(-\frac{2}{3}, 0\right) \).
Overall, the graph of \( y = |3x| - 2 \) is a V-shape that opens upwards, with its vertex at \( (0, -2) \) and intercepts at \( \left(\frac{2}{3}, 0\right) \) and \( \left(-\frac{2}{3}, 0\right) \).
1. **Absolute Value Function**: The term \( |3x| \) represents an absolute value function, which creates a V shape. The expression \( |3x| \) is equal to \( 3x \) when \( x \) is positive and \( -3x \) when \( x \) is negative.
2. **Vertical Shift**: The entire graph is shifted downward by 2 units due to the \(-2\) at the end of the equation. This means that the vertex (the point where the V opens) will be located at the point where \( |3x| = 0\), which occurs when \( x = 0 \). Thus, the vertex will be at the point \( (0, -2) \).
3. **Slope**: The slopes of the lines forming the V shape are determined by the coefficient of \( x \) in the absolute value function:
- For \( x \geq 0 \) (the right side of the V), the equation simplifies to \( y = 3x - 2 \).
- For \( x < 0 \) (the left side of the V), the equation simplifies to \( y = -3x - 2 \).
4. **Intercepts**:
- **Y-Intercept**: Substitute \( x = 0 \) into the equation:
\[
y = |3(0)| - 2 = 0 - 2 = -2.
\]
So the y-intercept is at \( (0, -2) \).
- **X-Intercepts**: Set \( y = 0 \):
\[
0 = |3x| - 2 \\
|3x| = 2.
\]
This leads to two equations:
\[
3x = 2 \implies x = \frac{2}{3}
\]
and
\[
3x = -2 \implies x = -\frac{2}{3}.
\]
So the x-intercepts are at \( \left(\frac{2}{3}, 0\right) \) and \( \left(-\frac{2}{3}, 0\right) \).
Overall, the graph of \( y = |3x| - 2 \) is a V-shape that opens upwards, with its vertex at \( (0, -2) \) and intercepts at \( \left(\frac{2}{3}, 0\right) \) and \( \left(-\frac{2}{3}, 0\right) \).