Question

choose the function types that are both increasing and decreasing for the same function.
select all that apply.

linear functions

constant functions

quadratic functions

exponential functions

linear absolute value functions

Explanation:

Step1: Analyze Linear Functions

Linear functions have the form \( y = mx + b \). The slope \( m \) is constant. If \( m>0 \), it's always increasing; if \( m<0 \), always decreasing. So it can't be both.

Step2: Analyze Constant Functions

Constant functions have \( y = c \) (constant). The function neither increases nor decreases, so it can't be both.

Step3: Analyze Quadratic Functions

Quadratic functions have the form \( y = ax^2 + bx + c \) (\( a
eq0 \)). The parabola opens up (if \( a>0 \)) or down (if \( a<0 \)). For \( a>0 \), it decreases on \( (-\infty, -\frac{b}{2a}) \) and increases on \( (-\frac{b}{2a}, \infty) \). For \( a<0 \), it increases on \( (-\infty, -\frac{b}{2a}) \) and decreases on \( (-\frac{b}{2a}, \infty) \). So it can be both.

Step4: Analyze Exponential Functions

Exponential functions: \( y = a^x \) (\( a>0, a
eq1 \)). If \( a>1 \), always increasing; if \( 0 < a < 1 \), always decreasing. Can't be both.

Step5: Analyze Linear Absolute Value Functions

Linear absolute value functions: \( y = |mx + b| \). The graph is a V - shape. For example, \( y = |x| \) decreases on \( (-\infty, 0) \) and increases on \( (0, \infty) \). So it can be both.

Answer:

Quadratic functions, Linear absolute value functions