3. Given that a quadratic function passes through (1,3)and (5,3)and has a negative y-intercept, which of the following could be the vertex of the function?



(4,-18)

(3,18)

(3,3)

(2,0)

We are given that a quadratic function passes through the points \( (1, 3) \) and \( (5, 3) \) and has a negative y-intercept. We need to determine which of the given points could be the vertex of the quadratic function.

### Step-by-step analysis:

1. **Symmetry of a quadratic function**:  
   A quadratic function has a parabolic shape and is symmetric about its vertex. If the function passes through two points with the same \( y \)-value, such as \( (1, 3) \) and \( (5, 3) \), then the vertex must lie exactly halfway between these two points.

2. **Midpoint of \( (1, 3) \) and \( (5, 3) \)**:  
   The x-coordinate of the vertex is the midpoint of the x-coordinates of these two points:
   \[
   x_{\text{vertex}} = \frac{1 + 5}{2} = \frac{6}{2} = 3
   \]
   So, the x-coordinate of the vertex is \( 3 \).

3. **Checking the y-coordinate of the vertex**:  
   The y-coordinate of the vertex could be different from the \( y \)-coordinates of the points \( (1, 3) \) and \( (5, 3) \). The vertex must have an x-coordinate of 3, which limits the possible options to those points where \( x = 3 \). From the given options:
   - \( (3, 18) \) has \( x = 3 \)
   - \( (3, 3) \) has \( x = 3 \)

4. **Determining the correct y-coordinate**:  
   Since the function passes through \( (1, 3) \) and \( (5, 3) \), and has a negative y-intercept, the parabola opens downwards. Thus, the vertex must be the highest point on the graph. The vertex of a downward-opening parabola is at its maximum, so the correct y-coordinate should be greater than 3.

   Among the two options where \( x = 3 \):
   - \( (3, 18) \) could be the vertex since 18 is greater than 3.
   - \( (3, 3) \) cannot be the vertex because the y-value is not greater than the points at \( (1, 3) \) and \( (5, 3) \).

### Conclusion:
The vertex of the quadratic function could be \( (3, 18) \).

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Would you like further details or have any questions?

Here are 5 related questions to deepen understanding:
1. How can you determine if a quadratic function opens upwards or downwards?
2. What is the significance of the axis of symmetry in a quadratic function?
3. How can you derive the equation of a quadratic function given its vertex and a point?
4. What is the vertex form of a quadratic equation, and how does it help in identifying the vertex?
5. How can the discriminant of a quadratic equation help determine the nature of its roots?

**Tip**: The vertex form of a quadratic equation is a helpful tool to quickly identify the vertex of the parabola. It is written as \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex.

Given that a quadratic function passes through (1,3) and (5,3) and has a negative y-intercept, which of the following could be the vertex of the function? (4,-18), (3,18), (3,3), (2,0)

Explore how to find the vertex of a quadratic function that passes through (1,3) and (5,3) with a negative y-intercept. This problem covers quadratic functions, symmetry, and the vertex form of a parabola. Perfect for students in Grades 9-10.

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