https://r2.erweima.ai/mathbot/image/1726624900906ib4itcm7.jpg

The image shows several quadratic functions \( F(x) \) and asks to find their intersections with the coordinate axes (both the x-axis and y-axis). Let's go through the process of solving these types of problems.

For each function:
- **To find the x-intercepts** (points where the graph crosses the x-axis), set \( F(x) = 0 \) and solve for \( x \).
- **To find the y-intercept** (the point where the graph crosses the y-axis), substitute \( x = 0 \) and solve for \( F(0) \).

Let's solve them one by one.

### a) \( F(x) = x^2 - 2x - 3 \)
- **x-intercepts**: Set \( F(x) = 0 \)
  \[
  x^2 - 2x - 3 = 0
  \]
  Factoring:
  \[
  (x - 3)(x + 1) = 0
  \]
  So, the solutions are \( x = 3 \) and \( x = -1 \).
  - x-intercepts: \( (-1, 0) \) and \( (3, 0) \)

- **y-intercept**: Set \( x = 0 \)
  \[
  F(0) = 0^2 - 2(0) - 3 = -3
  \]
  - y-intercept: \( (0, -3) \)

### b) \( F(x) = -x^2 + 6x - 8 \)
- **x-intercepts**: Set \( F(x) = 0 \)
  \[
  -x^2 + 6x - 8 = 0
  \]
  Multiply through by -1:
  \[
  x^2 - 6x + 8 = 0
  \]
  Factoring:
  \[
  (x - 2)(x - 4) = 0
  \]
  So, \( x = 2 \) and \( x = 4 \).
  - x-intercepts: \( (2, 0) \) and \( (4, 0) \)

- **y-intercept**: Set \( x = 0 \)
  \[
  F(0) = -(0)^2 + 6(0) - 8 = -8
  \]
  - y-intercept: \( (0, -8) \)

### c) \( F(x) = x^2 - 6x + 9 \)
- **x-intercepts**: Set \( F(x) = 0 \)
  \[
  x^2 - 6x + 9 = 0
  \]
  Factoring:
  \[
  (x - 3)^2 = 0
  \]
  So, \( x = 3 \).
  - x-intercept: \( (3, 0) \) (This is a double root, so it touches the x-axis at one point.)

- **y-intercept**: Set \( x = 0 \)
  \[
  F(0) = 0^2 - 6(0) + 9 = 9
  \]
  - y-intercept: \( (0, 9) \)

### d) \( F(x) = x^2 - 4x + 4 \)
- **x-intercepts**: Set \( F(x) = 0 \)
  \[
  x^2 - 4x + 4 = 0
  \]
  Factoring:
  \[
  (x - 2)^2 = 0
  \]
  So, \( x = 2 \).
  - x-intercept: \( (2, 0) \) (This is also a double root.)

- **y-intercept**: Set \( x = 0 \)
  \[
  F(0) = 0^2 - 4(0) + 4 = 4
  \]
  - y-intercept: \( (0, 4) \)

### e) \( F(x) = -x^2 - x + 6 \)
- **x-intercepts**: Set \( F(x) = 0 \)
  \[
  -x^2 - x + 6 = 0
  \]
  Multiply through by -1:
  \[
  x^2 + x - 6 = 0
  \]
  Factoring:
  \[
  (x - 2)(x + 3) = 0
  \]
  So, \( x = 2 \) and \( x = -3 \).
  - x-intercepts: \( (2, 0) \) and \( (-3, 0) \)

- **y-intercept**: Set \( x = 0 \)
  \[
  F(0) = -(0)^2 - (0) + 6 = 6
  \]
  - y-intercept: \( (0, 6) \)

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Would you like more details or have any questions about these solutions? Here are five related questions to explore:

1. What happens to the graph of a quadratic function when its discriminant is zero?
2. How does the sign of the leading coefficient affect the direction of the parabola?
3. What is the vertex form of a quadratic function, and how does it help in finding the vertex directly?
4. How can you find the axis of symmetry of a quadratic function from its general form?
5. How do the solutions of a quadratic function relate to the factorization method?

**Tip:** When solving quadratic equations, if factoring is difficult, you can always use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Determine the points of intersection of the graphs with the coordinate axes for the given quadratic functions.

Learn how to determine the points of intersection of quadratic functions with the x-axis and y-axis. Step-by-step explanations provided for finding x-intercepts and y-intercepts using quadratic equations. Suitable for students in Grades 8-10.

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