Which of the following functions has at least one vertex? Select all that apply.
A.f(x)=x+9

B.g(x)=2x2

C.b(x)=3x

D.m(x)=−x3

E.j(x)=∣x∣


A vertex in a function is typically the highest or lowest point on the graph of a quadratic function (parabola) or an absolute value function. Let's analyze each of the given functions to determine which ones have a vertex.

- **A. \( f(x) = x + 9 \)**:  
  This is a linear function, which graphs as a straight line. A straight line does not have a vertex, as there is no highest or lowest point.  
  **No vertex**.

- **B. \( g(x) = 2x^2 \)**:  
  This is a quadratic function (a parabola). Quadratic functions always have a vertex, which is either a minimum (if the parabola opens upwards) or a maximum (if it opens downwards).  
  **Has a vertex**.

- **C. \( b(x) = 3x \)**:  
  This is also a linear function, similar to option A.  
  **No vertex**.

- **D. \( m(x) = -x^3 \)**:  
  This is a cubic function. Cubic functions do not have a single highest or lowest point but rather have a point of inflection.  
  **No vertex**.

- **E. \( j(x) = |x| \)**:  
  This is an absolute value function. Absolute value functions graph as a "V" shape, and the point at the bottom of the "V" is the vertex.  
  **Has a vertex**.

### Functions with a vertex:
- **B. \( g(x) = 2x^2 \)**
- **E. \( j(x) = |x| \)**

Would you like more details or have any other questions?

### Related Questions:
1. What are the general forms of functions that always have a vertex?
2. How do you find the vertex of a quadratic function?
3. What is the significance of the vertex in the graph of an absolute value function?
4. Can a cubic function ever have a vertex, and why or why not?
5. How do transformations affect the vertex of a quadratic or absolute value function?

### Tip:
The vertex form of a quadratic function \( y = a(x-h)^2 + k \) directly reveals the vertex as the point \((h, k)\).

Which of the following functions has at least one vertex? Select all that apply.
A. f(x) = x + 9
B. g(x) = 2x^2
C. b(x) = 3x
D. m(x) = -x^3
E. j(x) = |x|

Learn which functions have vertices, including quadratic, absolute value, linear, and cubic functions. Understand how to identify a vertex in a graph with step-by-step analysis. Suitable for students in Grades 8-10.

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