Question

what kind of transformation converts the graph of $f(x) = -8x + 7$ into the graph of $g(x) = -8x - 1$? translation 1 unit down translation 1 unit up translation 1 unit right translation 1 unit left

Explanation:

Step1: Recall the vertical translation formula

For a function \(y = f(x)\), a vertical translation \(y=f(x)+k\) occurs. If \(k>0\), it's a translation up; if \(k < 0\), it's a translation down.

Step2: Rewrite the functions in the translation formula form

We have \(f(x)=-8x + 7\) and \(g(x)=-8x-1\). We can rewrite \(g(x)\) as \(g(x)=f(x)+(-8)\). But wait, let's use another approach. The general form of a linear function is \(y = mx + b\). For \(f(x)=-8x + 7\) (where \(b_1 = 7\)) and \(g(x)=-8x-1\) (where \(b_2=-1\)). The change in the \(y -\)intercept is \(\Delta b=b_2 - b_1\).
Calculate \(\Delta b=-1-7=-8\). Wait, no, another way: \(g(x)=f(x)-8\). But if we consider the transformation rule for \(y = f(x)\) to \(y=f(x)+k\). Let's check the difference in the constant terms.
We know that \(g(x)=f(x)-8\). But wait, if we use the formula for vertical shift: \(y = f(x)+k\). Here \(f(x)=-8x + 7\) and \(g(x)=-8x-1\). We can write \(g(x)=( - 8x + 7)-8\).

Answer:

translation 8 units down. But since there is a mistake in the problem - maybe a typo (if we assume the intended \(g(x)=-8x + 6\) (1 unit down: \(7-1 = 6\)) or if we consider the options given in a wrong - input - data scenario (assuming the intended \(g(x)=-8x+6\)):
If we assume the problem has a typo and the intended \(g(x)=-8x + 6\) (because among the given options, the formula for vertical shift \(y=f(x)+k\), with \(f(x)=-8x + 7\) and if \(k=-1\) (i.e., \(y=-8x+7-1=-8x + 6\)):

Step1: Apply vertical shift formula

For \(y = f(x)\) and \(y=f(x)+k\), when \(k=-1\) (since \(g(x)=f(x)-1\) where \(f(x)=-8x + 7\) and \(g(x)=-8x+6\) (assuming typo)).
The transformation is a translation 1 unit down.