find the vertex and y intercept. tell if the parabola opens up or down. find the y intercept y=x^2+6x+1



Let's solve the quadratic equation \( y = x^2 + 6x + 1 \) to find:

1. **The vertex**: The vertex form of a quadratic equation is \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex.
   
   The formula for the x-coordinate of the vertex is \( h = -\frac{b}{2a} \), where \( a \) and \( b \) come from the standard form of the quadratic equation \( y = ax^2 + bx + c \).
   
   Here, \( a = 1 \), \( b = 6 \), and \( c = 1 \).

   - The x-coordinate of the vertex is:
   \[
   h = -\frac{6}{2(1)} = -3
   \]

   - To find the y-coordinate, substitute \( x = -3 \) back into the equation:
   \[
   y = (-3)^2 + 6(-3) + 1 = 9 - 18 + 1 = -8
   \]

   So, the vertex is \( (-3, -8) \).

2. **The y-intercept**: The y-intercept occurs when \( x = 0 \).

   Substitute \( x = 0 \) into the equation:
   \[
   y = (0)^2 + 6(0) + 1 = 1
   \]

   So, the y-intercept is \( (0, 1) \).

3. **Opening of the parabola**: The sign of \( a \) (the coefficient of \( x^2 \)) tells us whether the parabola opens up or down.
   
   - Since \( a = 1 \), which is positive, the parabola opens **up**.

### Summary:
- **Vertex**: \( (-3, -8) \)
- **Y-intercept**: \( (0, 1) \)
- **Parabola opens up**.

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you find the x-intercepts of a quadratic equation?
2. What does the vertex tell you about the graph of a parabola?
3. How does the value of \( a \) affect the width of the parabola?
4. What happens to the graph if you change the constant \( c \) in the equation?
5. How do you complete the square to convert a quadratic equation into vertex form?

**Tip**: To quickly determine the direction of a parabola, just check the sign of the coefficient of \( x^2 \). Positive means it opens up, negative means it opens down.

find the vertex and y intercept. tell if the parabola opens up or down. find the y intercept y=x^2+6x+1

Learn how to find the vertex and y-intercept of the quadratic equation y = x^2 + 6x + 1, and determine whether the parabola opens up or down. This detailed solution covers key algebraic concepts, including the vertex formula and standard quadratic equations. Suitable for students in Grades 9-10.

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