Question

The vertex of the parabola $y=x^{2}-2x+4$ is shifted $p$ units to the right and then $q$ units up. If the resulting point is $(4,5)$, then the values of $p$ and $q$ respectively are

• A. 2 and 3
• B. 3 and 5
• C. 5 and 2
• D. 3 and 2
• E. 1 and 2

Hint:

Algebra Tip: You can also find the vertex instantly by completing the square! $y = x^2 - 2x + 4 \implies y = (x^2 - 2x + 1) + 3 \implies y = (x-1)^2 + 3$. The vertex $(h,k)$ is visible right away as $(1, 3)$.

Answer 1

The Correct Option is D

Concept:
The vertex of a parabola given in the standard form $y = ax^2 + bx + c$ can be found using the formula $x = -\frac{b}{2a}$ to find the x-coordinate, and then plugging it back into the equation to find the y-coordinate. A geometric shift "right" adds to the x-coordinate, and a shift "up" adds to the y-coordinate.

Step 1: Find the x-coordinate of the initial vertex.
For the parabola $y = x^2 - 2x + 4$, we have $a = 1$ and $b = -2$. $$x_{initial} = -\frac{-2}{2(1)}$$ $$x_{initial} = \frac{2}{2} = 1$$

Step 2: Find the y-coordinate of the initial vertex.
Substitute $x = 1$ back into the parabola's equation: $$y_{initial} = (1)^2 - 2(1) + 4$$ $$y_{initial} = 1 - 2 + 4 = 3$$ The initial vertex is $(1, 3)$.

Step 3: Apply the geometric shifts.
The vertex is shifted $p$ units to the right (adding to $x$) and $q$ units up (adding to $y$). The new vertex coordinates are conceptually $(1 + p, 3 + q)$.

Step 4: Equate to the final resulting point.
We are given that the final resulting point is $(4, 5)$. Set the coordinate pairs equal to each other: $$1 + p = 4$$ $$3 + q = 5$$

Step 5: Solve for p and q.
For p: $p = 4 - 1 \implies p = 3$ For q: $q = 5 - 3 \implies q = 2$ The values are 3 and 2 respectively. Hence the correct answer is (D) 3 and 2.