Question

Which of these equations has a vertex with the same vertical shift as \(y=(x-3)^2+1\) but with a vertex shifted to the left of the graph of this equation? Select all that apply.

a. \(y=(x-3)^2+5\)
b. \(y=(x-3)^2-2\)
c. \(y=(x-2)^2+1\)
d. \(y=(x-5)^2+1\)

Answer 1

Only equation c. y=(x-2)^2+1 has a vertex with the same vertical shift as y=(x-3)^2+1 and is shifted to the left, with the vertex at (2,1). Options a and b have different vertical shifts, and option d is shifted to the right.

Step-by-step explanation:

To determine which of the given equations has a vertex with the same vertical shift as y=(x-3)^2+1 but is shifted to the left, we need to compare the equations and identify changes in their vertex form. The vertex form of a quadratic equation is y = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola.

The equation y=(x-3)^2+1 has a vertex at (3,1). To match the vertical shift, we need the 'k' value to be the same (+1). So, we can immediately eliminate options a. y=(x-3)^2+5 and b. y=(x-3)^2-2 because their 'k' values are different (+5 and -2, respectively).

Now, for the parabola to shift to the left, the 'h' value needs to be less than 3. Looking at c. y=(x-2)^2+1 and d. y=(x-5)^2+1, we can see that option c has a vertex at (2,1), making it shifted one unit to the left of the original parabola's vertex, and option d has a vertex at (5,1), making it shifted two units to the right. Thus, only option c matches the criteria for the question.