To determine whether the given function [tex]\( f(x) = 9x + 2 \)[/tex] is linear, quadratic, or exponential, let's analyze the form and characteristics of each type of function.
### Linear Functions
A linear function has the form:
[tex]\[ f(x) = mx + b \][/tex]
where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants.
- The graph of a linear function is a straight line.
- The slope [tex]\( m \)[/tex] indicates the steepness and direction of the line.
- The constant [tex]\( b \)[/tex] represents the y-intercept, where the line crosses the y-axis.
### Quadratic Functions
A quadratic function has the form:
[tex]\[ f(x) = ax^2 + bx + c \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants.
- The graph of a quadratic function is a parabola.
- The coefficient [tex]\( a \)[/tex] determines whether the parabola opens upwards (if [tex]\( a > 0 \)[/tex]) or downwards (if [tex]\( a < 0 \)[/tex]).
### Exponential Functions
An exponential function has the form:
[tex]\[ f(x) = a \cdot b^x \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, with [tex]\( b > 0 \)[/tex] and [tex]\( b \neq 1 \)[/tex].
- The graph of an exponential function is a curve that shows exponential growth or decay.
- The base [tex]\( b \)[/tex] indicates the rate of growth (if [tex]\( b > 1 \)[/tex]) or decay (if [tex]\( 0 < b < 1 \)[/tex]).
### Analysis of the Given Function
The given function is:
[tex]\[ f(x) = 9x + 2 \][/tex]
- This function is in the form [tex]\( mx + b \)[/tex], where [tex]\( m = 9 \)[/tex] and [tex]\( b = 2 \)[/tex].
- It does not contain a term with [tex]\( x^2 \)[/tex] (which would make it quadratic) or [tex]\( b^x \)[/tex] (which would make it exponential).
- The highest degree of the variable [tex]\( x \)[/tex] is 1, characteristic of linear functions.
Therefore, the given function [tex]\( f(x) = 9x + 2 \)[/tex] is linear.
