Which equation best represents the graph shown below? Explain in detail how you arrived at your answer by stating each of the mathematical transformations necessary to produce the graph on your scratch work.
ik the answer is Y=(X+2)^(2)-3 I just don't know how to solve it
y=−(x−2)2−3
To determine which equation best represents the graph shown, we can analyze the mathematical transformations necessary to produce the graph.
First, let's start with the basic graph of y = x^2. This is a U-shaped graph that opens upwards and has the vertex at the origin (0, 0).
Now, let's go through each transformation step:
1. The graph is multiplied by -1: This reflects the graph across the x-axis, so the U-shape becomes an inverted U-shape.
2. The graph is shifted 2 units to the right: This means every x-value is increased by 2, shifting the entire graph to the right. The vertex, which was at (0, 0), will now be at (2, 0).
3. The graph is shifted 3 units downwards: This means every y-value is decreased by 3, shifting the entire graph downwards. The vertex, which was at (2, 0), will now be at (2, -3).
Putting all these transformations together, we have:
y = -(x-2)^2 - 3
This equation satisfies all the transformations described above and matches the graph shown.
To determine the equation that represents the given graph, we can analyze the mathematical transformations present on the graph.
1. The graph is a result of a reflection across the x-axis. This means that the original parabola opens upwards, but the given graph opens downwards. Since the negative sign is placed outside the equation, it indicates this reflection.
2. The vertex of the graph is at (-2, -3). This means that the parabola is shifted 2 units to the left and 3 units downwards.
Combining these two transformations, we can conclude that the equation representing the graph is of the form y = -((x + a)^2) + b.
Plugging in the values for the transformation, we have a = -2 and b = -3. Therefore, the equation becomes:
y = -(x + 2)^2 - 3
Thus, the equation that best represents the given graph is y = -(x + 2)^2 - 3.