11. Comparing Functions in Different Forms

GED® questions often ask students to compare two functions presented in different forms: an equation, a table, a graph, or a verbal description.

To compare functions, look for:

• Initial value: the output when \(x=0\), often the y-intercept.
• Rate of change: the slope for a linear function.
• Specific output: the value of the function at a named input.

Example: Function A is \(A(x)=2x+5\). Function B is shown by the table \((0,3),(1,7),(2,11)\).

Function A has initial value 5 and rate of change 2. Function B has initial value 3. Its output increases by 4 each time \(x\) increases by 1, so its rate of change is 4.

Therefore, Function A starts higher, but Function B grows faster.

Common misconception: comparing only one output value. A function with the larger value at \(x=0\) may not stay larger if the other function has a greater slope.

Academic habit: compare both starting value and rate of change before writing a conclusion.