Understanding how different types of functions relate to the shapes of their graphs is key to knowing how functions work on a coordinate plane. Each function type has its own special form, which makes its graph look different.
Here are the main types of functions:
Linear Functions: These are shown by the equation ( y = mx + b ). Here, ( m ) is the slope (how steep the line is) and ( b ) is where the line crosses the y-axis. The graphs of linear functions are straight lines that change at a steady rate.
Quadratic Functions: These are seen in equations like ( y = ax^2 + bx + c ). Their graphs form parabolas, which can open up or down depending on the value of ( a ). The highest or lowest point of the parabola is called the vertex.
Cubic Functions: These have the equation format ( y = ax^3 + bx^2 + cx + d ). Their graphs usually have one or two turns and create an S-like shape. They can cross the x-axis up to three times.
Exponential Functions: They are written in forms like ( y = ab^x ), where ( a ) is a constant and ( b ) is the base. Exponential functions grow (or shrink) quickly, and their graphs show a curve that gets very close to the x-axis but never actually touches it. This behavior is called asymptotic.
Absolute Value Functions: These are shown as ( y = |x| ). They create V-shaped graphs, which means the output is always positive or zero.
Knowing these connections helps us see how changes in a function’s equation affect its graph. This, in turn, influences the solutions to the equations and inequalities shown by these graphs.
Understanding how different types of functions relate to the shapes of their graphs is key to knowing how functions work on a coordinate plane. Each function type has its own special form, which makes its graph look different.
Here are the main types of functions:
Linear Functions: These are shown by the equation ( y = mx + b ). Here, ( m ) is the slope (how steep the line is) and ( b ) is where the line crosses the y-axis. The graphs of linear functions are straight lines that change at a steady rate.
Quadratic Functions: These are seen in equations like ( y = ax^2 + bx + c ). Their graphs form parabolas, which can open up or down depending on the value of ( a ). The highest or lowest point of the parabola is called the vertex.
Cubic Functions: These have the equation format ( y = ax^3 + bx^2 + cx + d ). Their graphs usually have one or two turns and create an S-like shape. They can cross the x-axis up to three times.
Exponential Functions: They are written in forms like ( y = ab^x ), where ( a ) is a constant and ( b ) is the base. Exponential functions grow (or shrink) quickly, and their graphs show a curve that gets very close to the x-axis but never actually touches it. This behavior is called asymptotic.
Absolute Value Functions: These are shown as ( y = |x| ). They create V-shaped graphs, which means the output is always positive or zero.
Knowing these connections helps us see how changes in a function’s equation affect its graph. This, in turn, influences the solutions to the equations and inequalities shown by these graphs.