Question

The parabola y= x^2 is changed to the form y= a(x-p)^2 +q by translating the parabola 3 units up and 4 units left and expanding it vertically by a factor of 2. What are the values of a, p, q?

Answer 1

The transformed equation is y = 2(x-4)² + 3, where a = 2, p = 4, and q = 3.

Step-by-step explanation:

When a parabola described by the equation y = x² is vertically stretched by a factor of 2, translated 3 units up and 4 units to the left, the new equation takes the form y = a(x-p)² + q. Here, the value of a represents the vertical stretch, so a = 2. The translation of 4 units to the left means that the vertex moves from (0,0) to (4,0), thus p = 4. Finally, translating the graph 3 units up indicates that the new vertex will be at (4,3), so q = 3. Therefore, the transformed equation is y = 2(x-4)² + 3.

Answer 2

The values for the transformed parabola equation y = a(x - p)^2 + q after translating 4 units left, 3 units up, and vertically expanding by a factor of 2 are a = 2, p = 4, and q = 3.

Step-by-step explanation:

The parabola y = x^2 is being transformed, so let's find the new values of a, p, and q.

Since the parabola is translated 4 units left, this affects the p value of the vertex form, moving it from 0 to 4 (in positive direction because left translation is positive in the vertex form). Therefore we have p = 4.

The translation 3 units up affects the q value, moving the vertex from 0 to 3 on the y-axis. So, q = 3.

The vertical expansion by a factor of 2 means that the a value is multiplied by 2. Since the original a was 1 (because y = x^2 is the same as y = 1·(x-0)^2 + 0), our new a is 2.

Therefore, the transformed equation is y = 2(x - 4)^2 + 3.