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 of a function is a point where the direction of the graph changes, typically associated with quadratic, absolute value, or higher-degree polynomial functions. Let’s examine each function to see if it has a vertex:

- **A. \( f(x) = x + 9 \):**  
  This is a linear function. Linear functions do not have vertices, as their graphs are straight lines.  
  **No vertex.**

- **B. \( g(x) = 2x^2 \):**  
  This is a quadratic function. Quadratic functions are parabolas, and every parabola has a vertex (the minimum or maximum point).  
  **Has a vertex.**

- **C. \( b(x) = 3x \):**  
  This is another linear function. As with option A, it does not have a vertex.  
  **No vertex.**

- **D. \( m(x) = -x^3 \):**  
  This is a cubic function. Cubic functions generally have points of inflection, but not vertices, as their graphs do not have a sharp turn or change direction at a single point.  
  **No vertex.**

- **E. \( j(x) = |x| \):**  
  This is an absolute value function, which forms a V-shaped graph. The point where the graph changes direction is called a 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?

Here are 5 related questions:
1. How do you find the vertex of a quadratic function?
2. What is the significance of the vertex in absolute value functions?
3. Can a cubic function ever have a vertex-like point?
4. How do you determine the direction of a parabola's opening?
5. How do transformations affect the vertex of an absolute value function?

**Tip:** For quadratic functions \( ax^2 + bx + c \), the vertex can be found using \( x = -\frac{b}{2a} \).

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 how to identify which functions have a vertex, focusing on quadratic and absolute value functions. This problem explains key concepts such as linear, quadratic, and cubic functions, and includes a step-by-step solution for determining the vertex. Suitable for students in Grades 8-10.

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