To analyze the function \( y = |3x| - 2 \), we need to identify its key characteristics, such as intercepts and the overall shape of the graph.
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Understanding the Function:
- The absolute value function \( |3x| \) will return non-negative values for all \( x \). As such, the graph will have a V-shape.
- The function equals zero when \( |3x| = 2 \), which means \( 3x = 2 \) or \( 3x = -2 \). This gives two \( x \)-intercepts:
- \( x = \frac{2}{3} \) (approximately 0.67)
- \( x = -\frac{2}{3} \) (approximately -0.67)
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Y-Intercept:
- To find the y-intercept (where \( x = 0 \)): \( y = |3(0)| - 2 = 0 - 2 = -2 \)
- Thus, the y-intercept is \( (0, -2) \).
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Shape of the Graph:
- The graph of \( y = |3x| - 2 \) will have a minimum (the vertex of the V-shape) at \( (0, -2) \) and will open upwards (the arms of the V go infinitely up).
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X-Intercepts:
- As calculated, the x-intercepts are \( (0.67, 0) \) and \( (-0.67, 0) \).
Now, let’s analyze the options provided against our findings:
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"a straight line with a positive slope has a y-intercept of (0, -2) and an x-intercept of (0.75, 0)": This does not match our graph since it mentions a straight line, not the V shape of an absolute value function.
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"a straight line with a positive slope has no y-intercept and an x-intercept of (2, 0)": This is incorrect as it incorrectly describes an aspect of the absolute value graph.
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"an angled line with a minimum value of (0, negative), which is also the y-intercept, has x-intercepts of (negative 0.75, 0) and (0.75, 0)": This might seem close, but the intercepts are incorrectly placed (they should be around \( \pm 0.67 \) not \( \pm 0.75 \)).
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"an angled line with a minimum value of (0.75, 0), has a y-intercept of (0, 2)": This is completely incorrect based on our earlier calculations.
In conclusion, none of the options provided accurately represent the graph of \( y = |3x| - 2 \). The graph indeed has a minimum at \( (0, -2) \) and x-intercepts at \( (-0.67, 0) \) and \( (0.67, 0) \), resembling a “V” shape opening upwards. The expected choices don’t match these characteristics.
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