Big Ideas Math Algebra 1, 2015
BI
Big Ideas Math Algebra 1, 2015 View details
7. Piecewise Functions
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Exercise 12 Page 221

Practice makes perfect
a We are told that the reference beam originates at and reflects off a mirror at Let's graph this situation.
Illustration of the reference beam
To write this as an absolute value function, the point of reflection is going to be the vertex. Recall the vertex form of an absolute value function.
Here the coordinate is the vertex. Let's substitute our vertex into this equation.
Now we can solve for by substituting the other known point, into the equation.
Solve for
We can now write our final equation.
b To write the absolute value function as a piecewise function, we first need to determine where the function changes direction as this is where our domains will split. In this case, this occurs when
Illustration of the reference beam. A vertical dashed line passes through (5,4)
When we find the domain for the left piece we keep in mind that the function begins at
The possible values for the right piece of the function are all real numbers greater than or equal to
Next, we can write the function as two separate equations by splitting our absolute value into its two cases, one positive and one negative.
We can simplify these equations a little bit. Let's start with the negative case.
Now we can simplify the positive case.
Finally, we can write the equation of the piecewise function.

Extra

Alternative Method
Rather than using the absolute value equation to solve for the piecewise function, as the example in the textbook showed, we can write the equations using the slope-intercept form.
We can use the graph of the absolute value function to find the slope of the left piece and the slope of the right piece. Then we use points from the graph, such as to solve for If done correctly, the answer will be the same.