Draw the graph of the quadratic function `x^(2)-4x+3` and hence find the roots of the equation `x^(2)-4x+3=0`. What is the minimum value of the function ?

Answer

To solve the problem step-by-step, we will first find the roots of the quadratic function \( f(x) = x^2 - 4x + 3 \), then we will draw the graph of the function, and finally, we will determine the minimum value of the function.

### Step 1: Finding the roots of the quadratic equation

1. **Identify the coefficients**:
The given quadratic function is \( f(x) = x^2 - 4x + 3 \).
Here, \( a = 1 \), \( b = -4 \), and \( c = 3 \).

2. **Calculate the discriminant**:
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values:
\[ D = (-4)^2 - 4 \cdot 1 \cdot 3 = 16 - 12 = 4 \]

3. **Determine the nature of the roots**:
Since \( D > 0 \), the roots are real and distinct.

4. **Find the roots using the quadratic formula**:
The roots can be found using the formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values:
\[ x = \frac{-(-4) \pm \sqrt{4}}{2 \cdot 1} = \frac{4 \pm 2}{2} \]
This gives us two roots:
\[ x_1 = \frac{6}{2} = 3 \quad \text{and} \quad x_2 = \frac{2}{2} = 1 \]

### Step 2: Drawing the graph of the quadratic function

1. **Identify the vertex of the parabola**:
The vertex \( (h, k) \) of the parabola can be found using:
\[ h = -\frac{b}{2a} = -\frac{-4}{2 \cdot 1} = 2 \]
To find \( k \), substitute \( h \) back into the function:
\[ k = f(2) = 2^2 - 4 \cdot 2 + 3 = 4 - 8 + 3 = -1 \]
Thus, the vertex is \( (2, -1) \).

2. **Plot the points**:
- The vertex \( (2, -1) \).
- The roots \( (1, 0) \) and \( (3, 0) \).
- Calculate another point, for example, at \( x = 0 \):
\[ f(0) = 0^2 - 4 \cdot 0 + 3 = 3 \quad \text{(point (0, 3))} \]

3. **Draw the graph**:
- Plot the points \( (1, 0) \), \( (2, -1) \), \( (3, 0) \), and \( (0, 3) \).
- Sketch the parabola opening upwards through these points.

### Step 3: Finding the minimum value of the function

The minimum value of the function occurs at the vertex, which we found to be \( (2, -1) \). Therefore, the minimum value of the function \( f(x) = x^2 - 4x + 3 \) is:
\[ \text{Minimum value} = -1 \]

### Summary of Results
- The roots of the equation \( x^2 - 4x + 3 = 0 \) are \( x = 1 \) and \( x = 3 \).
- The minimum value of the function is \( -1 \).