Question

given f(x) and g(x)=f(x + k), use the graph to determine the value of k.

Explanation:

Step1: Identify y - intercepts

The y - intercept of \(f(x)\) is 4 (where the red line crosses the y - axis), and the y - intercept of \(g(x)\) is 9.

Step2: Use transformation formula

We know that \(g(x)=f(x + k)\). A horizontal shift of the function \(y = f(x)\) to \(y=f(x + k)\) is related to the change in the y - intercepts for linear functions. The change in the y - intercepts gives us information about the horizontal shift. The formula for a horizontal shift of a linear function \(y = mx + b\) to \(y=m(x + k)+b\). The change in the y - intercepts of two parallel linear functions \(f(x)\) and \(g(x)=f(x + k)\) is related to the shift. Since \(g(x)\) is a horizontal shift of \(f(x)\), and the y - intercept of \(g(x)\) is 5 units above the y - intercept of \(f(x)\) for parallel lines (same slope), and for a left - shift of a linear function \(y = f(x)\) to \(y = f(x + k)\) (\(k>0\)), we can also use the fact that if we consider a point \((0,y_1)\) on \(f(x)\) and the corresponding point \((-k,y_2)\) on \(g(x)\) (because \(g(x)=f(x + k)\)). For linear functions with the same slope, the change in the y - values of the y - intercepts is related to the horizontal shift. The value of \(k\) is 5.

Answer:

5