To analyze the graph of the function f(x) = x² – 8x – 5, we can start by rewriting it in vertex form or by identifying key characteristics such as the vertex, axis of symmetry, and intercepts.
1. **Vertex**: The vertex of a quadratic function in standard form f(x) = ax² + bx + c can be found using the formula x = -b/(2a). In our case, a = 1 and b = -8, so:
x = -(-8)/(2*1) = 8/2 = 4.
To find the y-coordinate of the vertex, we substitute x back into the function:
f(4) = 4² – 8(4) – 5 = 16 – 32 – 5 = -21.
Therefore, the vertex is at the point (4, -21).
2. **Axis of Symmetry**: The axis of symmetry for a parabola is the vertical line that passes through the vertex. Here, the line is x = 4.
3. **Direction of Opening**: Since the coefficient of x² (which is 1) is positive, the parabola opens upwards.
4. **Y-Intercept**: To find the y-intercept, substitute x = 0 into the function:
f(0) = 0² – 8(0) – 5 = -5.
The y-intercept is at (0, -5).
5. **X-Intercepts**: To find the x-intercepts, we can solve the equation f(x) = 0:
x² – 8x – 5 = 0.
Using the quadratic formula x = [-b ± √(b² – 4ac)]/(2a), we find:
x = [8 ± √((-8)² – 4(1)(-5))]/(2*1) = [8 ± √(64 + 20)]/2 = [8 ± √84]/2 = [8 ± 2√21]/2 = 4 ± √21.
This shows there are two distinct x-intercepts.
In conclusion, the characteristics of the graph are:
- The vertex is at (4, -21).
- The parabola opens upwards.
- The axis of symmetry is x = 4.
- The y-intercept is (0, -5).
- There are two x-intercepts at (4 – √21, 0) and (4 + √21, 0).
By checking the statements regarding the graph, you can determine which ones are true based on the characteristics we’ve discussed.