Question

which linear function is represented by the graph? \\( f(x) = -2x + 1 \\) \\( f(x) = -\frac{1}{2}x + 1 \\) \\( f(x) = \frac{1}{2}x + 1 \\) \\( f(x) = 2x + 1 \\)

Explanation:

Step1: Recall slope-intercept form

The slope - intercept form of a linear function is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. From the graph, we can see that the line crosses the y - axis at \((0,1)\), so \(b = 1\).

Step2: Calculate the slope

We can use two points on the line to calculate the slope. Let's use the points \((0,1)\) and \((4,-1)\). The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Substituting \(x_1 = 0,y_1 = 1,x_2=4,y_2=-1\) into the formula, we get \(m=\frac{-1 - 1}{4-0}=\frac{-2}{4}=-\frac{1}{2}\).

Step3: Write the linear function

Since \(m =-\frac{1}{2}\) and \(b = 1\), the linear function is \(f(x)=-\frac{1}{2}x + 1\). We can also check with another point, for example, when \(x=-4\), \(f(-4)=-\frac{1}{2}\times(-4)+1=2 + 1=3\), which matches the point \((-4,3)\) on the graph.

Answer:

\(f(x)=-\frac{1}{2}x + 1\) (the option: \(f(x)=-\frac{1}{2}x + 1\))