Let's go through each part step by step for the given quadratic function:

The function is:

$y = x^2 + 8x + 15$

Part (a): Determine if the parabola opens upward or downward.

• The coefficient of $x^2$ is positive (it is 1), which means the parabola opens upward.

Answer: The parabola opens upward.

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Part (b): Find the vertex.

To find the vertex of a quadratic function in standard form $y = ax^2 + bx + c$, use the formula for the x-coordinate of the vertex: $x_{\text{vertex}} = \frac{-b}{2a}$ For $y = x^2 + 8x + 15$, we have:

• $a = 1$
• $b = 8$

So, $x_{\text{vertex}} = \frac{-8}{2(1)} = \frac{-8}{2} = -4$

Now, substitute $x = -4$ into the original equation to find the y-coordinate of the vertex: $y = (-4)^2 + 8(-4) + 15 = 16 - 32 + 15 = -1$

Thus, the vertex is at $(-4, -1)$.

Answer: The vertex is $(-4, -1)$.

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Part (c): Find the x-intercepts.

The x-intercepts occur when $y = 0$. Set the equation equal to 0: $0 = x^2 + 8x + 15$ Now, solve the quadratic equation. We can factor it: $x^2 + 8x + 15 = (x + 3)(x + 5) = 0$ Setting each factor equal to zero: $x + 3 = 0 \quad \text{or} \quad x + 5 = 0$ $x = -3 \quad \text{or} \quad x = -5$

Thus, the x-intercepts are $x = -3$ and $x = -5$.

Answer: The x-intercepts are $(-3, 0)$ and $(-5, 0)$.

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Part (d): Find the y-intercept.

The y-intercept occurs when $x = 0$. Substitute $x = 0$ into the equation: $y = (0)^2 + 8(0) + 15 = 15$

Thus, the y-intercept is $15$.

Answer: The y-intercept is $15$.

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Part (e): Use the results to graph the quadratic function.

• Vertex: $(-4, -1)$
• x-intercepts: $(-3, 0)$ and $(-5, 0)$
• y-intercept: $(0, 15)$
• The parabola opens upward (since $a > 0$).

With this information, you can plot these key points and sketch the curve of the parabola.

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Would you like to see more details or need help with graphing the function? Here are a few additional questions for you:

1. How do you factor a quadratic equation like $x^2 + 8x + 15$?
2. What other methods could you use to find the roots of a quadratic equation besides factoring?
3. How does the sign of $a$ in the quadratic equation affect the direction of the parabola?
4. Can you identify the axis of symmetry for the quadratic equation $y = x^2 + 8x + 15$?
5. What are the properties of the vertex in a parabola?

Tip: When finding the vertex, always remember that the formula $x = \frac{-b}{2a}$ gives you the axis of symmetry of the parabola!