To find the equation of the axis of symmetry and the coordinates of the vertex for the function given:

$y = x^2 - 4x - 3$

1. Equation of the Axis of Symmetry

For a quadratic function in the form $y = ax^2 + bx + c$, the axis of symmetry can be found using the formula: $x = -\frac{b}{2a}$ In this case:

• $a = 1$
• $b = -4$

Substitute these values into the formula: $x = -\frac{-4}{2 \times 1} = \frac{4}{2} = 2$

So, the equation of the axis of symmetry is: $x = 2$

2. Coordinates of the Vertex

The vertex lies on the axis of symmetry. To find the y-coordinate, substitute $x = 2$ back into the equation:

$y = (2)^2 - 4 \cdot 2 - 3$ $y = 4 - 8 - 3 = -7$

Thus, the coordinates of the vertex are: $(2, -7)$

Final Answer:

• The equation of the axis of symmetry is $x = 2$.
• The coordinates of the vertex are $(2, -7)$.

Would you like further details or have any other questions?

Here are some related questions you might consider:

1. How would the vertex change if the equation were $y = x^2 + 4x - 3$?
2. What is the axis of symmetry for the function $y = 2x^2 - 4x - 3$?
3. How can you tell the direction of the parabola's opening from the equation $y = x^2 - 4x - 3$?
4. How would you find the y-intercept of $y = x^2 - 4x - 3$?
5. What effect does changing the constant term have on the vertex?

Tip: The vertex formula is a quick way to find the maximum or minimum point of a parabola without graphing.