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To analyze the quadratic equation provided:

\[
y = x^2 - 8x + 7
\]

we need to determine if the parabola opens up or down, find whether it has a maximum or minimum, and identify the vertex and \(y\)-intercept.

### Step 1: Determine if the Parabola Opens Up or Down
In the equation \(y = ax^2 + bx + c\), the parabola's direction depends on the coefficient \(a\):
- If \(a > 0\), the parabola opens **upward** and has a **minimum** point.
- If \(a < 0\), it opens **downward** and has a **maximum** point.

Here, \(a = 1\) (positive), so the parabola opens **upward** and has a **minimum** point.

### Step 2: Find the Vertex
The vertex of a parabola given by \(y = ax^2 + bx + c\) is found using the formula:
\[
x = -\frac{b}{2a}
\]

For our equation:
- \(a = 1\)
- \(b = -8\)

Calculating \(x\)-coordinate of the vertex:
\[
x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4
\]

Substitute \(x = 4\) into the equation to find the \(y\)-coordinate:
\[
y = (4)^2 - 8 \cdot 4 + 7 = 16 - 32 + 7 = -9
\]

Thus, the vertex is at \((4, -9)\).

### Step 3: Find the \(y\)-Intercept
The \(y\)-intercept occurs where \(x = 0\). Substitute \(x = 0\) into the equation:
\[
y = (0)^2 - 8 \cdot 0 + 7 = 7
\]

So, the \(y\)-intercept is \((0, 7)\).

### Answers
1. **The graph opens up and has a minimum**.
2. **The vertex is \((4, -9)\)**.
3. **The \(y\)-intercept is \((0, 7)\)**.

Would you like more details on any of these steps, or do you have any questions?

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Here are five related questions to expand on this topic:

1. How do you determine the axis of symmetry of a parabola?
2. What effect does the coefficient \(c\) have on the graph of a parabola?
3. How would the graph change if the coefficient \(a\) were negative?
4. How do you find the roots of a quadratic equation like this one?
5. What is the relationship between the vertex form of a quadratic equation and the standard form?

**Tip:** For any quadratic equation, knowing whether \(a\) is positive or negative instantly tells you if the parabola opens up or down, which helps in identifying if it has a maximum or minimum.

Consider the parabola given by the equation: y = x^2 - 8x + 7. Does the graph open up or down? Does the graph have a maximum or a minimum? The vertex is ( , ). The y-intercept is (0, ).

This problem explains how to analyze the quadratic function y = x^2 - 8x + 7. Learn whether the graph opens up or down, find the vertex, determine if there's a maximum or minimum point, and calculate the y-intercept. Suitable for Grades 8-10 students studying quadratic functions.

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