Líst the key features of the quadratic function graphed.
A) x-intercepts: (0,0), y-intercept: (0,10), vertex (3,-1)
B) x-intercepts: none, y-intercept: (0,-10), vertex: (3,-1)
C) x-intercepts: (0.-10), y-intercept: none, vertex: (3,-1)
D) x-intercepts: none, y’intercept: (3,-1), vertex (0-10)
The Correct Answer and Explanation is :
The correct answer is B) x-intercepts: none, y-intercept: (0, -10), vertex: (3, -1).
Explanation:
A quadratic function is typically in the form (y = ax^2 + bx + c), and its graph is a parabola. Key features of a quadratic function include the x-intercepts, y-intercept, and vertex.
- x-intercepts: These are the points where the graph crosses the x-axis (where (y = 0)). If there are no x-intercepts, the parabola does not touch or cross the x-axis, indicating that the quadratic equation does not have real solutions. In this case, the graph has no x-intercepts (as specified in option B).
- y-intercept: This is the point where the graph crosses the y-axis (where (x = 0)). The y-intercept is found by plugging (x = 0) into the quadratic equation and solving for (y). In option B, the y-intercept is ((0, -10)), meaning that when (x = 0), the value of (y) is (-10).
- Vertex: The vertex is the highest or lowest point on the graph, depending on the direction the parabola opens (upwards for a positive (a), downwards for a negative (a)). The vertex in option B is ((3, -1)), meaning the point ((3, -1)) is the turning point of the parabola, which opens upwards because the vertex represents the minimum point.
Why Option B is Correct:
- The x-intercepts in option B are listed as “none,” which means the parabola does not cross the x-axis. This suggests the equation has no real solutions or the graph does not intersect the x-axis.
- The y-intercept is ((0, -10)), which matches the description in option B. This is where the graph intersects the y-axis, and it is consistent with the given data.
- The vertex at ((3, -1)) represents the lowest point of the parabola, confirming that the graph opens upward.
Why Other Options are Incorrect:
- Option A has an incorrect y-intercept and vertex. The y-intercept is ((0, 10)), but it should be ((0, -10)), and the x-intercepts are also wrongly listed.
- Option C suggests the x-intercepts are ((0, -10)), but that is incorrect because (x)-intercepts should not have a y-value of -10. Additionally, there is no y-intercept in this option.
- Option D lists the vertex as ((3, -1)), but the y-intercept is incorrectly shown as ((3, -10)), and this is inconsistent with the vertex and overall graph characteristics.
Thus, option B is the best description of the given quadratic function.