1. **Problem Statement:** Sketch the graph of the function $y = (x + 1)(x + 3)$. 2. **Formula and Important Rules:** The function is a quadratic in factored form. To sketch it, find the roots (x-intercepts), the y-intercept, and the vertex (turning point). The general form is $y = (x - r_1)(x - r_2)$ where $r_1$ and $r_2$ are roots. 3. **Find the x-intercepts:** Set $y=0$ to find roots. $$0 = (x + 1)(x + 3)$$ So, $x + 1 = 0$ or $x + 3 = 0$ $$x = -1 \quad \text{or} \quad x = -3$$ These are the points $(-1, 0)$ and $(-3, 0)$. 4. **Find the y-intercept:** Set $x=0$ $$y = (0 + 1)(0 + 3) = 1 \times 3 = 3$$ So the y-intercept is $(0, 3)$. 5. **Find the vertex (turning point):** The vertex lies midway between the roots. $$x_{vertex} = \frac{-1 + (-3)}{2} = \frac{-4}{2} = -2$$ Calculate $y$ at $x = -2$: $$y = (-2 + 1)(-2 + 3) = (-1)(1) = -1$$ So the vertex is at $(-2, -1)$. Since the coefficient of $x^2$ is positive (1), the parabola opens upwards and the vertex is a minimum point. 6. **Summary of key points:** - Roots: $(-1, 0)$ and $(-3, 0)$ - Y-intercept: $(0, 3)$ - Vertex: $(-2, -1)$ (minimum point) 7. **Sketching the graph:** Plot the roots on the x-axis, the y-intercept on the y-axis, and the vertex below the x-axis at $(-2, -1)$. Draw a smooth parabola opening upwards through these points.