Graphing Linear Functions

Problem: Graph the linear function \( f \) given by \( f(x) = 2x + 4 \).

Solution:

• You need only two points to graph a linear function. These points may be chosen as the \( x \)- and \( y \)-intercepts of the graph.
• Determine the \( x \)-intercept: Set \( f(x) = 0 \) and solve for \( x \): \[ 2x + 4 = 0 \implies 2x = -4 \implies x = -2 \] Giving the point \((-2, 0)\).
• Determine the \( y \)-intercept: Set \( x = 0 \) to find \( f(0) \): \[ f(0) = 2(0) + 4 = 4 \] Giving the point \((0, 4)\).
• The graph of the above function is a line passing through the points \((-2, 0)\) and \((0, 4)\) as shown below:

  

Problem: Graph the linear function \( f \) given by \( f(x) = x + 3 \).

Problem: Graph the linear function \( f \) given by \( f(x) = -\left(\frac{1}{3}\right)x - \frac{1}{2} \).

Solution:

• Determine the \( x \)-intercept: Set \( f(x) = 0 \) and solve for \( x \): \[ -\frac{1}{3}x - \frac{1}{2} = 0 \implies -\frac{1}{3}x = \frac{1}{2} \implies x = -\frac{3}{2} \] Giving the point \(\left(-\frac{3}{2}, 0\right)\).
• Determine the \( y \)-intercept: Set \( x = 0 \) to find \( f(0) \): \[ f(0) = -\frac{1}{2} \] Giving the point \(\left(0, -\frac{1}{2}\right)\).
• The graph of the above function is a line passing through the points \(\left(-\frac{3}{2}, 0\right)\) and \(\left(0, -\frac{1}{2}\right)\) as shown below:

  

Problem: Graph the linear function \( f \) given by \( f(x) = -\frac{x}{5} + \frac{1}{3} \).