Question

Determine the vertex and axis of symmetry of y = x^2 - 10x + 3.
vertex: (___); axis of symmetry: x = ____

Answer 1

To determine the vertex and axis of symmetry of the quadratic function \( y = x^2 - 10x + 3 \), we can use the vertex formula.

The standard form of a quadratic equation is given as \( y = ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = -10 \)
- \( c = 3 \)

The x-coordinate of the vertex can be found using the formula:

\[
x = -\frac{b}{2a}
\]

Substituting the values of \( a \) and \( b \):

\[
x = -\frac{-10}{2 \cdot 1} = \frac{10}{2} = 5
\]

Next, we can find the y-coordinate of the vertex by substituting \( x = 5 \) back into the original equation:

\[
y = (5)^2 - 10(5) + 3
\]
\[
y = 25 - 50 + 3
\]
\[
y = -22
\]

Therefore, the vertex is \( (5, -22) \).

The axis of symmetry is the vertical line that goes through the vertex. Since the x-coordinate of the vertex is 5, the axis of symmetry is:

\[
x = 5
\]

In conclusion, we have:

- Vertex: \( (5, -22) \)
- Axis of symmetry: \( x = 5 \)

So your answers are:
- Vertex: \( (5, -22) \)
- Axis of symmetry: \( x = 5 \)