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?