Question

Identify the graph of f(x) = x² - 8x + 15

Answer 1

To analyze the graph of the quadratic function \( f(x) = x^2 - 8x + 15 \), we can follow these steps:

1. **Identify the vertex**: The quadratic function is in the standard form \( f(x) = ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = -8 \)
- \( c = 15 \)

The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \):
\[
x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4
\]
Now, we can find the y-coordinate of the vertex by substituting \( x = 4 \) back into the function:
\[
f(4) = 4^2 - 8 \cdot 4 + 15 = 16 - 32 + 15 = -1
\]
Thus, the vertex is at the point \( (4, -1) \).

2. **Find the x-intercepts**: To find the x-intercepts, set \( f(x) = 0 \):
\[
x^2 - 8x + 15 = 0
\]
We can factor this quadratic:
\[
(x - 3)(x - 5) = 0
\]
This gives us the solutions:
\[
x = 3 \quad \text{and} \quad x = 5
\]
So the x-intercepts are at the points \( (3, 0) \) and \( (5, 0) \).

3. **Find the y-intercept**: To find the y-intercept, set \( x = 0 \):
\[
f(0) = 0^2 - 8 \cdot 0 + 15 = 15
\]
Thus, the y-intercept is at the point \( (0, 15) \).

4. **Direction of the parabola**: Since the coefficient \( a = 1 \) is positive, the parabola opens upwards.

Now, summarizing:

- The vertex is at \( (4, -1) \).
- The x-intercepts are \( (3, 0) \) and \( (5, 0) \).
- The y-intercept is \( (0, 15) \).
- The parabola opens upwards.

To visualize the graph:

- The parabola's vertex is the lowest point.
- It crosses the x-axis at \( (3, 0) \) and \( (5, 0) \).
- It crosses the y-axis at \( (0, 15) \).

Using this information, you can sketch the graph of \( f(x) = x^2 - 8x + 15 \) showing all the key points mentioned.