To analyze the quadratic function \( y = x^2 - 8x \), we will determine the key points: vertex, x-intercepts, and y-intercept.
-
Vertex:
The vertex of a quadratic in the form \( y = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:
\[ x = -\frac{b}{2a} \]
In this case, \( a = 1 \) and \( b = -8 \), so:
\[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \]
To find the y-coordinate, substitute \( x = 4 \) back into the equation:
\[ y = (4)^2 - 8(4) = 16 - 32 = -16 \]
Therefore, the vertex is \( (4, -16) \). -
X-Intercepts:
To find the x-intercepts, set \( y = 0 \):
\[ 0 = x^2 - 8x \]
Factor the equation:
\[ 0 = x(x - 8) \]
Thus, the solutions are \( x = 0 \) and \( x = 8 \).
Hence, the x-intercepts are \( (0, 0) \) and \( (8, 0) \). -
Y-Intercept:
To find the y-intercept, set \( x = 0 \):
\[ y = 0^2 - 8(0) = 0 \]
Therefore, the y-intercept is \( (0, 0) \).
Summary of Key Points:
- Vertex: \( (4, -16) \)
- X-intercepts: \( (0, 0) \) and \( (8, 0) \)
- Y-intercept: \( (0, 0) \)
Correct Response:
The correct responses corresponding to the analysis above are:
- x-intercept(s): \( (0, 0) \) and \( (8, 0) \)
- y-intercept: \( (0, 0) \)
- vertex: \( (4, -16) \)
Now, looking at the options you provided, this analysis would correspond to:
- x-intercepts: \( (0, 0) \) (not explicitly mentioned in any option, but included)
- vertex: \( (4, -16) \)
- y-intercept: \( (0, 0) \)
This matches the summary derived from the function.