https://file.aiquickdraw.com/mathbot/file/1728087725078as5ntvap.jpg

The function given in the image is:

\[
y = x^2 + 6x + 7
\]

This is a quadratic function, and its graph will be a parabola. Let's break down how to graph it:

### Steps to Graph:
1. **Identify the key features:**
   - **Vertex**: The vertex form of a quadratic function is \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex. You can find the vertex using the formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = 6 \), so:
     \[
     x = -\frac{6}{2(1)} = -3
     \]
     Now, substitute \( x = -3 \) into the equation to find the \( y \)-coordinate of the vertex:
     \[
     y = (-3)^2 + 6(-3) + 7 = 9 - 18 + 7 = -2
     \]
     The vertex is at \( (-3, -2) \).

2. **Y-intercept**: The y-intercept occurs when \( x = 0 \):
   \[
   y = 0^2 + 6(0) + 7 = 7
   \]
   So, the y-intercept is at \( (0, 7) \).

3. **Axis of Symmetry**: The axis of symmetry is the vertical line passing through the vertex, which is \( x = -3 \).

4. **Additional Points**: To accurately plot the graph, choose other values of \( x \), such as \( x = -2 \) and \( x = -4 \), and compute the corresponding \( y \)-values:
   - For \( x = -2 \), \( y = (-2)^2 + 6(-2) + 7 = 4 - 12 + 7 = -1 \).
   - For \( x = -4 \), \( y = (-4)^2 + 6(-4) + 7 = 16 - 24 + 7 = -1 \).

5. **Plot the points**: 
   - Vertex: \( (-3, -2) \)
   - Y-intercept: \( (0, 7) \)
   - Points: \( (-2, -1) \) and \( (-4, -1) \)

6. **Draw the parabola**: Since \( a = 1 \) (positive), the parabola opens upwards. Use the points to sketch a smooth curve.

Would you like further details or help with graphing? Here are some related questions:

1. How do you convert a quadratic equation to vertex form?
2. How do you determine whether a parabola opens upwards or downwards?
3. What is the significance of the axis of symmetry in a quadratic graph?
4. How do you find the roots of a quadratic equation?
5. How do changes in the coefficients of \( x^2 \), \( x \), and the constant affect the graph?

**Tip**: The vertex provides a quick way to know the minimum or maximum point of the parabola, depending on its orientation!

Graph the quadratic function y = x^2 + 6x + 7.

Learn how to graph the quadratic function y = x^2 + 6x + 7. This involves calculating the vertex, finding the y-intercept, and plotting key points to draw the parabola. Suitable for students in Grades 9-10.

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