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 \(g(x)\) in terms of \(f(x)\)

We have \(f(x)=-8x + 7\) and \(g(x)=-8x-1\). Notice that \(g(x)=f(x)-8\).
Let's check the \(y -\)intercepts. The \(y -\)intercept of \(f(x)\) is \(y = 7\) (when \(x = 0\), \(f(0)=-8\times0 + 7=7\)), and the \(y -\)intercept of \(g(x)\) is \(y=-1\) (when \(x = 0\), \(g(0)=-8\times0-1=-1\)). The change in the \(y -\)value from \(f(x)\) to \(g(x)\) is \(\Delta y=-1 - 7=-8\).

Answer:

translation 8 units down

(Note: There might be a mistake in the original problem's options as the correct vertical translation is 8 units down. If we assume it's a mis - typing and the functions are \(f(x)=-8x + 7\) and \(g(x)=-8x+6\) (translation 1 unit down) or \(f(x)=-8x + 8\) and \(g(x)=-8x - 1\) (translation 9 units down), but based on the given \(f(x)=-8x + 7\) and \(g(x)=-8x-1\), the vertical translation is 8 units down. If we consider the general form of linear functions \(y=mx + b\) and the vertical shift formula \(y_2=y_1 + k\), \(k=-8\))