Absolute Value Functions Essay, Research Paper
Absolute Value Functions (y=a|x-b|+c)
In an absolute value function “a” represents the slope of the equation. The slope is counted up or down, then both left and right thus producing the function’s “V” shape and its symmetry. If a*0 in the equation, then the “V” has and upward direction, but if a*0 then the “V” has a downward direction. This reflection is around y=c. “a” also helps determine the range of the function. If a*0 then y c, in other words the “V” will face downward and will have a range of all real numbers less than 0. If a*0 then y*c, or the “V” will have an upward direction and the range of all real numbers greater than 0. If a=0 then y=c, which means you will have a horizontal line with the range being whatever “c” may equal. “a” also represents the dilation, or size of the “V”. If |a|*1 then the “V” in turn is widened. If |a|*1 then the “V” becomes more narrow. If |a|=1 then the “V” retains its normal appearance with a growth rate of 1:1 on either side. If |a|=0 then the “V” becomes a horizontal line. If |a|=?then the “V” becomes a vertical ray, which is not really a function.
In absolute value functions, the letter “b” represents the x-translation, or where the vertex of the “V” is located along the x axis. In an x-translation, the “V” slides opposite the sign that is located within the absolute value bars. For example, if the equation reads y=|x-3| then the vertex will land on the positive number three on the right side of the y axis, if it reads y=|x+3| then the vertex will land on the negative number three on the left side of the y axis. The line of symmetry will be located where x=b.
Y-translations are represented by the letter “c” in absolute value functions. During a y-translation, the “V” slides up and down the y axis consistant with the sign of “c”. Therefore, if “c” is positive, the vertex of the “V” will be located in the positive half of the y axis, and if “c” is positive, then the vertex will be located in negative portion of the y axis.
The domain of absolute value functions are always all real numbers, for they are ever expanding. The vertex of these functions will be found at (b,c).