explore the significance of the
coefficient "a" in determining the shape of the graph of y=a|x|.[
To explore the significance of the coefficient "a" in determining the shape of the graph of y = a|x|, let's break down the equation and understand the impact of "a" on the graph.
The graph of y = a|x| represents an absolute value function, involving the absolute value of x. The absolute value function returns the distance of a number from zero, irrespective of its sign. Therefore, for any value of x, the expression |x| will always be positive or zero.
Now, let's analyze the role of the coefficient "a":
1. "a" determines the vertical stretch or compression:
When "a" is positive, the graph stretches vertically.
When "a" is negative, the graph becomes a reflection (a mirror image) across the x-axis, resulting in compression.
2. "a" changes the slope of the graph:
With a positive "a" value, the graph has a constant positive slope on both sides of the y-axis.
When "a" is negative, the graph has a constant negative slope on both sides of the y-axis.
3. "a" scales the amplitude:
The magnitude of "a" increases or decreases the amplitude (height) of the graph.
Let's consider some examples to further illustrate these effects:
- For y = 2|x|, the graph will have a steeper slope, and the height will be doubled compared to y = |x|.
- For y = -0.5|x|, the graph will be a mirror image across the x-axis, with the height reduced to half compared to y = |x|.
- For y = 3|x|, the graph will be three times as tall as the graph of y = |x|.
In summary, the coefficient "a" in y = a|x| significantly impacts the shape, slope, and amplitude of the graph. Positive values stretch the graph, while negative values cause reflection and compression. The magnitude of "a" scales the height of the graph.