Sketch the graph of f(x) = x^2 - 4x +9. The sketch must include the coordinates of any points where the graph of f(x) = x^2 - 4x +9 meets the coordinate axes.

To sketch the graph of the quadratic function \( f(x) = x^2 - 4x + 9 \), follow these steps:

### 1. **Find the Vertex:**
   The quadratic function is in the form \( f(x) = ax^2 + bx + c \). For this function:
   \[
   a = 1, \quad b = -4, \quad c = 9
   \]
   The x-coordinate of the vertex is given by:
   \[
   x = -\frac{b}{2a} = -\frac{-4}{2(1)} = \frac{4}{2} = 2
   \]
   Substituting \( x = 2 \) into the function to find the y-coordinate:
   \[
   f(2) = (2)^2 - 4(2) + 9 = 4 - 8 + 9 = 5
   \]
   So, the vertex is at \( (2, 5) \).

### 2. **Determine the Y-intercept:**
   The y-intercept is found by evaluating the function at \( x = 0 \):
   \[
   f(0) = (0)^2 - 4(0) + 9 = 9
   \]
   So, the graph meets the y-axis at \( (0, 9) \).

### 3. **Determine the X-intercepts:**
   The x-intercepts are found by solving the equation \( f(x) = 0 \):
   \[
   x^2 - 4x + 9 = 0
   \]
   The discriminant (\( b^2 - 4ac \)) for this quadratic is:
   \[
   (-4)^2 - 4(1)(9) = 16 - 36 = -20
   \]
   Since the discriminant is negative, the quadratic equation has no real solutions, meaning the graph does not intersect the x-axis.

### 4. **Sketch the Graph:**
   - The graph is a parabola opening upwards because \( a > 0 \).
   - The vertex is at \( (2, 5) \).
   - The graph crosses the y-axis at \( (0, 9) \).
   - There are no x-intercepts.

### Graphical Sketch:
- The parabola opens upwards with its vertex at \( (2, 5) \).
- The y-intercept is at \( (0, 9) \).
- The parabola is symmetric about the vertical line \( x = 2 \).

If you would like me to generate an actual visual representation of this sketch, I can do that too. Would you like to see the graph?

Sketch the graph of f(x) = x^2 - 4x + 9. The sketch must include the coordinates of any points where the graph of f(x) = x^2 - 4x + 9 meets the coordinate axes.

Learn how to graph the quadratic function f(x) = x^2 - 4x + 9 by finding its vertex, y-intercept, and analyzing its discriminant. Understand how to determine the graph's key points and sketch the parabola accurately. Suitable for students in Grades 9-10.

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