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.
(1 point)
Responses
y=(x+2)2−3
y is equal to open paren x plus 2 close paren squared minus 3
y=(x−2)2−3
y is equal to open paren x minus 2 close paren squared minus 3
y=12(x+2)2−3
y is equal to 1 half times open paren x plus 2 close paren squared minus 3
y=−(x−2)2−3
y is equal to negative open paren x minus 2 close paren squared minus 3
To determine which equation best represents the graph, we need to analyze the key features of the graph.
1. The graph is a parabola. This tells us that the equation will be of the form y = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola.
2. The vertex of the parabola is at the point (-2, -3). This means that the equation should have (x+2) and (x-2) terms to account for the horizontal shift.
3. The graph opens downwards, indicating a negative leading coefficient. This means that the equation should have a negative sign in front of the (x-2) term.
Taking these key features into consideration, the equation that best represents the graph is y = -(x-2)^2 - 3. This equation includes the necessary transformations: a horizontal shift to the left by 2 units (x-2) and a reflection over the x-axis (negative sign).
To determine which equation best represents the given graph, let's analyze the graph and identify the necessary mathematical transformations.
1. From the graph, we can see that the vertex is at point (-2, -3). This means that the vertex form of the equation will be (x-h)^2 + k, where (h, k) represents the vertex.
2. The graph is a downward-facing parabola, meaning the coefficient in front of the squared term will be negative.
3. Comparing the given equations, we can eliminate the options y=(x+2)^2-3 and y=12(x+2)^2-3 because they both have positive coefficients. The equation that remains is y=-(x-2)^2-3.
Now let's break down the mathematical transformations applied to the basic function y=x^2 to arrive at the equation y=-(x-2)^2-3:
- The negative sign in front of the function (-x^2) causes a reflection in the x-axis, resulting in the parabola opening downwards.
- The (+2) inside the parentheses (-(x-2)^2) shifts the parabola 2 units to the right. So, the vertex, which is normally at (0, 0), is now at (-2, 0).
- Finally, the (-3) outside the parentheses (-(x-2)^2-3) shifts the entire graph downward by 3 units, placing the vertex at (-2, -3) as observed in the graph.
Therefore, the equation y=-(x-2)^2-3 best represents the given graph.