Which of the following functions has at least one vertex? Select all that apply.

A.f(x)=x+9f(x)=x+9

B.g(x)=2x2g(x)=2x2

C.b(x)=3xb(x)=3x

D.m(x)=−x3m(x)=−x3

E.j(x)=∣x∣j(x)=∣x∣


To determine which functions have at least one vertex, we need to identify the types of functions listed:

A. \( f(x) = x + 9 \)  
   - This is a linear function. Linear functions do not have a vertex.

B. \( g(x) = 2x^2 \)  
   - This is a quadratic function. Quadratic functions have a vertex.

C. \( b(x) = 3x \)  
   - This is also a linear function, which does not have a vertex.

D. \( m(x) = -x^3 \)  
   - This is a cubic function. Cubic functions do not have a single vertex, though they may have inflection points.

E. \( j(x) = |x| \)  
   - This is an absolute value function. Absolute value functions have a vertex.

**The functions that have at least one vertex are:**

- **B.** \( g(x) = 2x^2 \)
- **E.** \( j(x) = |x| \) 

These are the correct selections.

Would you like further details or have any other questions?

Here are some related questions:
1. What is the vertex of a quadratic function?
2. How do you find the vertex of an absolute value function?
3. What is the general form of a cubic function?
4. Can a linear function have a vertex if it's a constant function?
5. What does the vertex represent in the context of real-world problems?

**Tip:** In a quadratic function, the vertex can be found using the formula \( x = -\frac{b}{2a} \), where the function is in the form \( ax^2 + bx + c \).

Learn which functions have at least one vertex with a detailed explanation of quadratic and absolute value functions. Explore examples such as f(x) = x + 9, g(x) = 2x^2, b(x) = 3x, m(x) = -x^3, and j(x) = |x|. Suitable for high school students studying functions and vertex analysis.

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