https://file.aiquickdraw.com/mathbot/file/1726801297009015qpmej.jpg

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.

Examine the graph of the quadratic function y = x^2 - 2x - 3 and answer several questions about the graph.

Learn how to analyze the quadratic function y = x^2 - 2x - 3. This includes finding the vertex, axis of symmetry, zeros, and more. Perfect for students in Grades 8-10 learning about parabolas and quadratic equations.

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