Question

Determine the vertex and axis of symmetry of y = x^2 - 10x + 3.

Answer 1

To determine the vertex and axis of symmetry of the quadratic function \( y = x^2 - 10x + 3 \), we can use the formula for the vertex of a parabola in the form \( y = ax^2 + bx + c \).

1. **Identifying coefficients**:
- Here, \( a = 1 \), \( b = -10 \), and \( c = 3 \).

2. **Finding the x-coordinate of the vertex**:
The x-coordinate of the vertex can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Plugging in the values of \( a \) and \( b \):
\[
x = -\frac{-10}{2 \cdot 1} = \frac{10}{2} = 5
\]

3. **Finding the y-coordinate of the vertex**:
To find the y-coordinate, substitute \( x = 5 \) back into the original equation:
\[
y = (5)^2 - 10(5) + 3 = 25 - 50 + 3 = -22
\]

Thus, the vertex is:
\[
(5, -22)
\]

4. **Finding the axis of symmetry**:
The axis of symmetry is a vertical line that passes through the vertex and can be found using the x-coordinate of the vertex. Therefore, the axis of symmetry is:
\[
x = 5
\]

**Summary**:
- The vertex of the parabola is \( (5, -22) \).
- The axis of symmetry is \( x = 5 \).