# Keep Your Eye On The Variable

The following equations all have a similarity:

y = |x – 8| + 5
y  = 4(x – 6) – 7
(x – 3)(13x + 11) = 0
y = (x + 1)^2 – 9

The similarity is that they all have expressions like (x- 6) or (13x + 11), which are often either translations or factors. Situations such as this occur with:
– Point-slope form of the equation of a line
– Vertex form of the equation of a quadratic
– Horizontal or vertical translation of any function
– Factors of a polynomial
– “Bounce point” of an absolute value function

I often find myself suggesting that students focus on the variable, not the constant. By asking “what value of the variable will make this expression equal zero?”, they will arrive at the correct conclusion.  Focusing directly on the constant will usually lead to sign errors (or worse).

Taking the above examples one at a time:

y = |x – 8| + 5
This is the equation of a linear absolute value function. What value of x will make the expression inside the absolute value signs equal to zero? Positive eight. Therefore, the graph of this equation will “bounce” at x = +8, and this can be considered a positive eight unit horizontal translation of the parent function y = |x|. Furthermore, if you subtract the five from both sides
(y – 5) = |x – 8|
you can now treat x and y identically. What value of y will make the expression on the left equal to zero? Positive five. Therefore, the graph of the parent function y = |x| has been translated (shifted) in the y-direction by positive five units (up).

y  = 4(x – 6) – 7
This is the equation of a line. What value of x will make the expression inside parentheses equal to zero? Positive six. Therefore, the graph of this equation’s parent function y = 4x has been translated (shifted) in the x-direction by positive six units (to the right). Furthermore, if you add the seven to both sides
(y + 7) = 4(x – 6)
you can now treat x and y identically. What value of y makes the expression inside the parentheses equal to zero? Negative seven. Therefore, the graph of this equation’s parent function y = 4x has also been translated (shifted) in the y-direction by negative seven units (down).

(x – 3)(13x + 11) = 0
This is a quadratic equation in factored form, set equal to zero. We can therefore rely on the “zero product property” to conclude that one or more of the factors must equal zero. Many students leap for the numbers (-3 and +11 in this case), to their eventual chagrin. I encourage students to always write out equations that set each of the factors equal to zero, then solve them:
x – 3 = 0
x = 3

13x + 11 = 0
13x = -11
x = -11/13
The values of x above are the two x-intercepts on the graph, the two values of x at which the result of the function is zero.

y = (x + 1)^2 – 9
This is a quadratic equation in vertex form.  I encourage students to re-write it so that both variables can be analyzed the same way:
(y + 9)  = (x + 1)^2
Asking “what value of y makes the expression on the left equal to zero” tells us that the parent function, y = x^2, has been translated vertically by negative nine. Asking the same question of the expression on the right tells us that the parent function has also been translated horizontally by negative one.

So, when faced with factors or translations, use algebra to move additive constants next to variables if necessary, then ask what value of the variable will make the expression equal zero.  That will be the value to use in your analysis. 