Let's break down and solve the problem from the graph given for the quadratic function $y = x^2 + 4x + 3$.

Part (a): Finding the Coordinates of the Turning Point

The turning point of a quadratic function $y = ax^2 + bx + c$ is given by the formula: $x = -\frac{b}{2a}$ For the equation $y = x^2 + 4x + 3$, we have:

• $a = 1$
• $b = 4$

Plugging in these values: $x = -\frac{4}{2 \cdot 1} = -2$

Now, substitute $x = -2$ back into the equation to find the $y$-coordinate: $y = (-2)^2 + 4(-2) + 3 = 4 - 8 + 3 = -1$

Thus, the coordinates of the turning point are: $(-2, -1)$

Part (b): Finding the Roots of the Equation $x^2 + 4x + 3 = 0$

To find the roots, we can factorize the quadratic equation: $x^2 + 4x + 3 = 0$ We look for two numbers that multiply to $3$ and add up to $4$. These numbers are $3$ and $1$, so we can rewrite the equation as: $(x + 3)(x + 1) = 0$

Setting each factor to zero gives: $x + 3 = 0 \Rightarrow x = -3$ $x + 1 = 0 \Rightarrow x = -1$

Therefore, the roots of the equation are: $(-3, 0) \text{ and } (-1, 0)$

Summary of Answers:

• Turning Point: $(-2, -1)$
• Roots: $(-3, 0)$ and $(-1, 0)$

Would you like a more detailed explanation or have any other questions?

Additional Questions for Practice:

1. How would you find the vertex of a quadratic equation in vertex form?
2. What is the axis of symmetry for the quadratic function $y = x^2 + 4x + 3$?
3. How can you determine if the turning point is a maximum or a minimum for any quadratic function?
4. How do the roots of a quadratic equation relate to its graph?
5. Can you solve $x^2 + 4x + 5 = 0$ by factoring or would another method be needed?

Quick Tip:

For any quadratic equation $ax^2 + bx + c = 0$, if it’s factorable, you can use the factors to quickly find the roots by setting each factor to zero.