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The problem you provided is written in Japanese and deals with a quadratic function in the form \( y = ax^2 + bx + c \). Here's the breakdown of the task:

1. The vertex of the quadratic function is \( (1, k) \), and it passes through the point \( (2, 2) \).
2. You are asked to fill in the blanks based on the given conditions.

Let's go through the questions:

### (1) For \( k = -2 \), determine \( a \), \( b \), and \( c \) in the equation.

### (2) The range of \( k \) such that the quadratic graph intersects the x-axis at two distinct points.

Let's solve each part systematically.

### Part (1): Solving for \( a \), \( b \), and \( c \) when \( k = -2 \)

Given:
- The quadratic equation has a vertex \( (1, k) \).
- The function passes through the point \( (2, 2) \).
  
For a parabola with vertex \( (1, k) \), the general form of the quadratic equation can be written as:

\[
y = a(x - 1)^2 + k
\]

When \( k = -2 \), this becomes:

\[
y = a(x - 1)^2 - 2
\]

We know it passes through \( (2, 2) \), so substituting \( x = 2 \) and \( y = 2 \) into the equation:

\[
2 = a(2 - 1)^2 - 2
\]
\[
2 = a(1)^2 - 2
\]
\[
2 = a - 2
\]
\[
a = 4
\]

Now we can rewrite the equation as:

\[
y = 4(x - 1)^2 - 2
\]

To expand this, we get:

\[
y = 4(x^2 - 2x + 1) - 2 = 4x^2 - 8x + 4 - 2 = 4x^2 - 8x + 2
\]

Thus, comparing this with the standard form \( y = ax^2 + bx + c \), we have:
- \( a = 4 \)
- \( b = -8 \)
- \( c = 2 \)

### Part (2): Finding the range of \( k \) for which the quadratic function intersects the x-axis at two distinct points

The general quadratic form is \( y = a(x - 1)^2 + k \), and we want to know for what range of \( k \) the graph intersects the x-axis at two distinct points.

The condition for two distinct real intersections is that the discriminant of the quadratic equation must be greater than zero. The discriminant of a quadratic function \( ax^2 + bx + c \) is given by:

\[
\Delta = b^2 - 4ac
\]

For this to have two distinct real roots, we require \( \Delta > 0 \). Here, solving this inequality will give us the range for \( k \).

---

Would you like me to go through the discriminant calculation step by step for Part (2)? 

Here are five related questions you might find useful:
1. What is the significance of the vertex form \( y = a(x - h)^2 + k \) in quadratic functions?
2. How does the value of \( a \) affect the shape of the parabola?
3. Why does the discriminant determine the number of real intersections with the x-axis?
4. How can we determine whether the quadratic function opens upwards or downwards?
5. How does shifting the vertex affect the graph of a parabola?

**Tip:** Always ensure that when solving quadratic equations, you check if the discriminant gives you real solutions (positive for two, zero for one, negative for none).

Consider the quadratic function y = ax^2 + bx + c, where the vertex is (1, k) and the function passes through the point (2, 2). Given that k = -2, find a, b, and c, and determine the range of k such that the quadratic function intersects the x-axis at two distinct points.

Learn how to find the quadratic equation given its vertex and a point, and determine the range for k such that the graph intersects the x-axis at two distinct points. The problem explores the vertex form and discriminant concepts, suitable for students in Grades 9-11.

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