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)=3^x

D.m(x)=−x^3

E.j(x)=∣x∣


To determine which of the functions have at least one vertex, let's analyze each one:

1. **A. \( f(x) = x + 9 \)**: This is a linear function, which is a straight line. Linear functions do not have vertices.

2. **B. \( g(x) = 2x^2 \)**: This is a quadratic function (a parabola). Quadratic functions always have a vertex, which is the maximum or minimum point of the parabola.

3. **C. \( b(x) = 3^x \)**: This is an exponential function. Exponential functions do not have vertices, as they do not have a maximum or minimum point like a parabola.

4. **D. \( m(x) = -x^3 \)**: This is a cubic function. Cubic functions do not have a single vertex; instead, they have an inflection point where the curve changes concavity.

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

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

Would you like more details or have any questions about these functions?

### Related Questions:
1. What is the vertex form of a quadratic function?
2. How do you find the vertex of a quadratic function?
3. Can a cubic function ever have a vertex?
4. What is the significance of a vertex in an absolute value function?
5. How does the graph of a linear function differ from that of a quadratic function?

### Tip:
The vertex of a quadratic function in the form \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).

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) = 3^x
D. m(x) = -x^3
E. j(x) = |x|

Explore how to determine which functions have at least one vertex. This detailed explanation covers quadratic and absolute value functions, providing insight into their properties and how to identify vertices. Perfect for students in Grades 9-11.

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