To find the vertex and axis of symmetry of the quadratic function \( y = x^2 - 10x + 3 \), we can use the vertex form of a parabola or the vertex formula.
The standard form of a quadratic function is \( y = ax^2 + bx + c \). For the given quadratic:
- \( a = 1 \)
- \( b = -10 \)
- \( c = 3 \)
Step 1: Finding the x-coordinate of the vertex
The x-coordinate of the vertex can be found using the formula:
\[ x = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \):
\[ x = -\frac{-10}{2 \cdot 1} = \frac{10}{2} = 5 \]
Step 2: Finding the y-coordinate of the vertex
To find the y-coordinate of the vertex, substitute \( x = 5 \) back into the original equation:
\[ y = (5)^2 - 10(5) + 3 \] \[ y = 25 - 50 + 3 \] \[ y = -22 \]
Conclusion
The vertex of the parabola is at the point \( (5, -22) \).
Step 3: Finding the axis of symmetry
The axis of symmetry for a quadratic function in standard form is given by the equation \( x = \) (the x-coordinate of the vertex). Thus, the axis of symmetry is:
\[ x = 5 \]
Final Answer
Vertex: \( (5, -22) \); Axis of symmetry: \( x = 5 \)
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