right parenthesis space equals space open vertical bar x close vertical bar is vertically compressed by a factor of 1 third and reflected across the x-axis, then the new equation is f left parenthesis x right parenthesis equals space minus 1 third open vertical bar x close vertical bar.
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f(x) = ...
To understand how the equation is transformed, let's break it down step by step:
1. Original Equation: f(x) = |x|
The original equation is f(x) = |x|, which represents the absolute value of x. This function outputs the distance of x from the origin (0) on a number line.
2. Vertical Compression:
When the equation is vertically compressed by a factor of 1/3, it means that the output values (y-values) of the function are compressed or reduced in height. We achieve this compression by multiplying the function by 1/3.
f(x) = (1/3)|x|
3. Reflection across the x-axis:
Reflecting the equation across the x-axis means that the entire function will be inverted or mirrored with respect to the x-axis. In other words, the positive and negative values of y will be switched. We achieve this reflection by multiplying the function by -1.
f(x) = -1(1/3)|x| = -1/3|x|
So, the new equation after the given transformations is:
f(x) = -1/3|x|
In summary, the original equation f(x) = |x| is vertically compressed by a factor of 1/3 and reflected across the x-axis to obtain the new equation f(x) = -1/3|x|.