Question

Graph the function f(x)=
–
8x2–1.

Answer 1

To graph the function \( f(x) = -8x^2 - 1 \), let's analyze its characteristics:

1. **Type of function**: This is a quadratic function, which is in the standard form \( f(x) = ax^2 + bx + c \). In this case, \( a = -8 \), \( b = 0 \), and \( c = -1 \).

2. **Direction of the parabola**: Since the coefficient \( a \) is negative (-8), the parabola opens downwards.

3. **Vertex**: The vertex of a parabola given by \( f(x) = ax^2 + bx + c \) can be found at the point \( (h, k) \), where
\[
h = -\frac{b}{2a} \quad \text{and} \quad k = f(h)
\]
Here, since \( b = 0 \):
\[
h = -\frac{0}{2(-8)} = 0
\]
To find \( k \):
\[
k = f(0) = -8(0^2) - 1 = -1
\]
Thus, the vertex is at \( (0, -1) \).

4. **Y-intercept**: The y-intercept occurs when \( x = 0 \):
\[
f(0) = -1
\]

5. **X-intercepts**: To find the x-intercepts, set \( f(x) \) to 0:
\[
0 = -8x^2 - 1 \implies 8x^2 = -1
\]
Since this results in no real solution (as \(-1\) does not equal a positive number), there are no x-intercepts.

6. **Additional Points**: We can plot a few more points to help shape the graph. For example:
- \( f(1) = -8(1^2) - 1 = -8 - 1 = -9 \)
- \( f(-1) = -8(-1^2) - 1 = -8 - 1 = -9 \)
- \( f(2) = -8(2^2) - 1 = -32 - 1 = -33 \)
- \( f(-2) = -8(-2^2) - 1 = -32 - 1 = -33 \)

### Summary of Key Points
- Vertex: \( (0, -1) \)
- Y-intercept: \( (0, -1) \)
- Points: \( (1, -9) \), \( (-1, -9) \), \( (2, -33) \), \( (-2, -33) \)

### Graph
To graph the function, plot the vertex at \( (0, -1) \) and additional points \( (1, -9) \), \( (-1, -9) \), \( (2, -33) \), and \( (-2, -33) \). Then, draw a downward-opening parabola through these points.

Here's a verbal description of what the graph would look like:

```
|
| *
| |
| |
| * |
| | |
| \ |
| \ |
| \ *
---|-------\--|---------
| \ |
| \|
|
|
```

This graph represents the function \( f(x) = -8x^2 - 1 \).