This week in Pre-calculus 11 we had our chapter 5 unit test and started unit 8. Something that I learned this week was how to graph the absolute value of both linear and quadratic functions, as well as how to write them in piecewise notation.

Recall:

• When finding the absolute value of something it is asking how far away a number is from zero. The absolute value will always be a positive number.

Need to know:

• The critical point (also called the point of inflection) is the point along the x-axis where the graph of the function changes direction.
• Piecewise notation, the first part of piecewise notation is used to state which parts of the original line are positive and didn’t need to be reflected, uses signs (≤ or ≥). The second part is used to describe the part of the function that had to be reflected upwards, uses signs (< or >).

Example 1:

• First, you graph the function as if there were no absolute value symbols around it. When graphing it normally we know that the y-intercept is 4 and that the linear function has a slope of 2.
• Next we graph the absolute value of the function. Since we already know that absolute values will always be positive we take the part of the function that runs past the x-axis and reflect it upwards. Reflecting it upwards will put it above the x-axis, making it positive (it’s the same as multiplying the negative points by -1). The part of the line that is already in the positive stays the same.
• After graphing the absolute value of the function you need to write the piecewise notation. first we write a notation for the portion of the function that stays the same. Since the right side of the graph doesn’t change we would write the notation as, 2x+4, x ≥ -2 (-2 is the critical point and doesn’t change). For the second part we write about the part of the graph that was reflected, when reflecting the values, it is the same as multiplying by -1. we would write the notation as -(2x+4), x < -2 (we only use less than because the critical point never changed).

Example 2:

•  Graph the function as if there were no absolute value symbols around it. When graphing it normally we know that the vertex is (3,-2) and that the quadratic function has a scale of 2-6-10.
• Next we graph the absolute value of the function. To do this we take the part of the parabola that is in the negative and reflect it upwards so that it is positive. To do this all you need to do is change the sign of the y in the vertex to a positive –> (3,2). The part of the function that is in the positive stays that same.
• After graphing the absolute value of the function you need to write the piecewise notation. First we write the notation about the part of the function that remained the same, we know that , x ≤ 2, x ≥ 4 (we write about the sides of the function since they didn’t change). We then write about the part of the function that was reflected upwards (multiplied by -1). We know that , 2 < x < 4 (in between those two points is where the function changes).