Let's analyze the problem given in the image. The task revolves around a quadratic function $y = x^2 - 2x - 3$ and involves answering several questions based on its graph.

The questions are:

a. How would you describe the graph?

This is a parabola, opening upwards. The shape is symmetric around the vertical axis passing through the vertex.

b. Give the vertex of the parabola and its axis of symmetry.

The vertex of a parabola given by $y = ax^2 + bx + c$ can be found using the formula for the x-coordinate of the vertex: $x = \frac{-b}{2a}$ Here, $a = 1$, $b = -2$, and $c = -3$. So: $x = \frac{-(-2)}{2(1)} = 1$ Substitute $x = 1$ into the equation to find the y-coordinate of the vertex: $y = 1^2 - 2(1) - 3 = 1 - 2 - 3 = -4$ Thus, the vertex is $(1, -4)$, and the axis of symmetry is the vertical line $x = 1$.

c. At what values of $x$ does the graph intersect the x-axis?

The x-intercepts can be found by solving $x^2 - 2x - 3 = 0$. Factoring the quadratic: $(x - 3)(x + 1) = 0$ Thus, the x-intercepts are $x = 3$ and $x = -1$.

d. What do you call these x-coordinates where the curve crosses the x-axis?

These x-coordinates are called the zeros or roots of the quadratic function.

e. What is the value of $y$ at these values of $x$?

At the x-intercepts, $y = 0$, since the graph crosses the x-axis at those points.

Would you like further details on any of these points?

Here are five related questions for further understanding:

1. What is the significance of the vertex in a quadratic graph?
2. How does the coefficient of $x^2$ (positive or negative) affect the direction of the parabola?
3. How do you determine if a quadratic function has two, one, or no x-intercepts?
4. Can the axis of symmetry ever be horizontal? Why or why not?
5. What other methods can be used to find the zeros of a quadratic function?

Tip: Remember, the discriminant $\Delta = b^2 - 4ac$ of the quadratic equation tells you how many real roots (zeros) the function has.