Question

the vertex is (4, - 9). a pair of symmetrical points are: (-9, 4). the x - intercepts are: 4. the y - intercept is: -9. the minimum y - value is: (image of a parabola on a coordinate grid with x from -10 to 10 and y from -10 to 10)

Explanation:

Vertex

Step1: Recall vertex definition

The vertex of a parabola is the point where it changes direction. For a parabola opening upwards (as this one does, since it has a minimum), the vertex is the lowest point. From the graph and the given box, we identify the vertex coordinates.

Step2: Confirm vertex

The graph shows the vertex at \((4, -9)\), which matches the boxed value. So the vertex is correctly \((4, -9)\).

Symmetrical Points

Step1: Recall symmetry of parabola

A parabola is symmetric about its axis of symmetry (the vertical line through the vertex, \(x = 4\) here). Symmetrical points should be equidistant from the axis of symmetry. The given point \((-9, 4)\) is not symmetric with respect to \(x = 4\). Let's find a correct pair. For example, if we take a point \((0, 10)\) (y - intercept), its symmetric point across \(x = 4\) would be \((8, 10)\) (since the distance from \(x = 0\) to \(x = 4\) is \(4\), so \(4 + 4 = 8\)). But the given box has \((-9, 4)\), which is incorrect. A correct pair could be \((0, 10)\) and \((8, 10)\) (or other points equidistant from \(x = 4\)).

x - intercepts

Step1: Recall x - intercept definition

x - intercepts are where the graph crosses the x - axis (where \(y = 0\)). From the graph, we can see the parabola crosses the x - axis at two points. Let's find their coordinates. The axis of symmetry is \(x = 4\). Let's assume one x - intercept is at \(x = 4 + d\) and the other at \(x = 4 - d\). From the graph, we can estimate the x - intercepts. Looking at the grid, when \(y = 0\), the x - values are around \(x=-1\) and \(x = 9\) (since the distance from \(x = 4\) to \(x = 9\) is \(5\), and to \(x=-1\) is also \(5\)). So the x - intercepts are \(x=-1\) and \(x = 9\), so the x - intercepts are \(-1\) and \(9\) (or the points \((-1, 0)\) and \((9, 0)\)). The given box has \(4\), which is incorrect ( \(x = 4\) is the axis of symmetry, not an x - intercept).

y - intercept

Answer:

s:

• Vertex: \(\boldsymbol{(4, -9)}\) (correct as per the vertex definition for the upward - opening parabola)
• Symmetrical Points: The given \((-9, 4)\) is incorrect. A correct pair could be \(\boldsymbol{(0, 10)}\) and \(\boldsymbol{(8, 10)}\) (or other points equidistant from \(x = 4\))
• x - intercepts: The given \(4\) is incorrect. The x - intercepts are \(\boldsymbol{-1}\) and \(\boldsymbol{9}\) (or the points \((-1, 0)\) and \((9, 0)\))
• y - intercept: The given \(-9\) is incorrect. The y - intercept is \(\boldsymbol{10}\) (the point \((0, 10)\))
• Minimum y - value: \(\boldsymbol{-9}\) (since the vertex \((4, -9)\) is the minimum point for the upward - opening parabola)