Let's solve each part of the problem for the quadratic function $f(x) = x^2 - 8x + 15$.

Given:

$f(x) = x^2 - 8x + 15$ This is a quadratic function in the form $ax^2 + bx + c$ where $a = 1$, $b = -8$, and $c = 15$.

Part (a): Determine whether the graph opens up or down.

• Since $a = 1$ (which is positive), the parabola opens up.
• So, the correct answer is: D. opens up because $a > 0$.

Part (b): Find the vertex.

The vertex of a quadratic function $f(x) = ax^2 + bx + c$ is given by: $h = -\frac{b}{2a} \quad \text{and} \quad k = f(h)$

1. Calculate $h$: $h = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4$
2. Calculate $k$: $k = f(4) = (4)^2 - 8 \cdot 4 + 15 = 16 - 32 + 15 = -1$

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

Part (c): Find the axis of symmetry.

The axis of symmetry for a parabola $f(x) = ax^2 + bx + c$ is the vertical line through the vertex: $x = h = 4$ So, the axis of symmetry is $x = 4$.

Part (d): Find the x-intercepts.

To find the x-intercepts, set $f(x) = 0$: $x^2 - 8x + 15 = 0$ This can be factored as: $(x - 3)(x - 5) = 0$ So, the x-intercepts are: $x = 3 \quad \text{and} \quad x = 5$ Therefore, the answer is A. The x-intercepts are $x = 3$ and $x = 5$.

Part (e): Find the y-intercept.

The y-intercept occurs when $x = 0$: $f(0) = 0^2 - 8 \cdot 0 + 15 = 15$ So, the y-intercept is 15.

Summary:

• (a) The parabola opens up because $a > 0$.
• (b) The vertex is (4, -1).
• (c) The axis of symmetry is $x = 4$.
• (d) The x-intercepts are $x = 3$ and $x = 5$.
• (e) The y-intercept is 15.

Would you like further explanation on any of these parts?

Follow-up Questions:

1. How do you find the vertex of a quadratic function in general?
2. What determines the direction in which a parabola opens?
3. How can you determine the number of x-intercepts a parabola has?
4. Why is the y-intercept given by substituting $x = 0$ into the function?
5. How does the axis of symmetry relate to the vertex of a parabola?

Tip:

For any quadratic function $f(x) = ax^2 + bx + c$, the sign of $a$ determines if it opens up (if $a > 0$) or down (if $a < 0$).