https://file.aiquickdraw.com/mathbot/file/1729543498532f7qqgfkq.jpg

The equation in the image is:

\[
y = x^2 - 3x + 8
\]

This is a quadratic equation, where the highest degree of \(x\) is 2, making it a parabola when graphed. Here are some key properties:

1. **Standard Form**: The equation is already in standard form, \( y = ax^2 + bx + c \), where:
   - \( a = 1 \) (coefficient of \(x^2\)),
   - \( b = -3 \) (coefficient of \(x\)),
   - \( c = 8 \) (constant term).

2. **Vertex**: The vertex of a parabola can be found using the formula for the x-coordinate:
   \[
   x_{\text{vertex}} = -\frac{b}{2a}
   \]
   Substituting the values:
   \[
   x_{\text{vertex}} = -\frac{-3}{2(1)} = \frac{3}{2} = 1.5
   \]
   To find the y-coordinate, substitute \(x = 1.5\) into the original equation:
   \[
   y_{\text{vertex}} = (1.5)^2 - 3(1.5) + 8 = 2.25 - 4.5 + 8 = 5.75
   \]
   So the vertex is \((1.5, 5.75)\).

3. **Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex, so it is \( x = 1.5 \).

4. **Y-intercept**: The y-intercept occurs when \(x = 0\). Substituting \(x = 0\) into the equation:
   \[
   y = 0^2 - 3(0) + 8 = 8
   \]
   So the y-intercept is \((0, 8)\).

5. **Direction of Opening**: Since the coefficient of \(x^2\) (i.e., \(a = 1\)) is positive, the parabola opens upwards.

Would you like to explore any specific detail or have further questions?

Here are 5 related questions for further exploration:
1. How would you find the x-intercepts of this quadratic function?
2. What is the discriminant of this quadratic equation, and how does it help determine the number of solutions?
3. How can we convert this equation into vertex form?
4. What does the graph of this function look like, and how can we sketch it?
5. How do transformations (like shifts or stretches) affect the graph of this parabola?

**Tip**: The discriminant of a quadratic equation helps determine how many real solutions (roots) it has.

Explore the quadratic equation y = x^2 - 3x + 8, including its graph, vertex, y-intercept, and axis of symmetry. Learn how to calculate the vertex and analyze the properties of the parabola. Suitable for students in Grades 8-10.

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