The problem asks us to find the zeros of the quadratic function $y = x^2 - 2x - 3$ by graphing. Here's how to solve it step-by-step:

Step 1: Understand Zeros of a Quadratic Function

The zeros (or roots) of a function are the $x$-values where the graph intersects the $x$-axis, meaning $y = 0$. For $y = x^2 - 2x - 3$, we are looking for the values of $x$ that make the equation $x^2 - 2x - 3 = 0$.

Step 2: Identify the Zeros from the Graph

From the graph provided:

• The parabola crosses the $x$-axis at $x = -1$ and $x = 3$.

Thus, the zeros of $y = x^2 - 2x - 3$ are: $x = -1 \quad \text{and} \quad x = 3$

Step 3: Verify the Results Algebraically (Optional)

The quadratic equation $x^2 - 2x - 3 = 0$ can be factored: $x^2 - 2x - 3 = (x - 3)(x + 1) = 0$ Setting each factor equal to zero: $x - 3 = 0 \quad \Rightarrow \quad x = 3$ $x + 1 = 0 \quad \Rightarrow \quad x = -1$ This confirms the zeros are $x = -1$ and $x = 3$.

Final Answer:

The zeros of the quadratic function $y = x^2 - 2x - 3$ are: $x = -1 \quad \text{and} \quad x = 3$

Would you like further explanation or details?

Relative Questions:

1. What are the steps to solve any quadratic equation by factoring?
2. How do you determine the vertex of a parabola from its equation?
3. Can all quadratic equations be factored? If not, what are the alternatives?
4. How does the discriminant of a quadratic equation relate to the graph?
5. What is the general shape of a parabola when the leading coefficient is positive?

Tip:

When solving quadratic equations graphically, always verify your results by substituting the zeros back into the equation to ensure $y = 0$.