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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 \( c = 15 \).

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#### 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**.

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### 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?

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#### 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 \( a < 0 \)).

For the function, f(x) = x^2 - 8x + 15, complete parts (a) through (e): (a) Determine whether the graph opens up or down, (b) Find the vertex, (c) Find the axis of symmetry, (d) Find the x-intercepts, if any, (e) Find the y-intercept.

Learn how to solve the quadratic function f(x) = x^2 - 8x + 15 with detailed steps. This problem covers finding the vertex, axis of symmetry, x-intercepts, and y-intercept, including how to determine the direction the parabola opens. Suitable for students in Grades 9-11.

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