Question

The parabola \(y=4-x^{2}\) has vertex \(P\). It intersects the \(x\)-axis at \(A\) and \(B\). If the parabola is translated from its initial position to a new position by moving its vertex along the line \(y=x+4\), so that it intersects the \(x\)-axis at \(B\) and \(C\), then the abscissa of \(C\) will be:

• A. 12
• B. 8
• C. 6
• D. $\frac{7}{3}$

Hint:

Parabolas are perfectly symmetrical! Since the original parabola has a width of 4 units (from $-2$ to $+2$), the translated parabola must also have a width of exactly 4 units on either side of its new vertex axis $x = 5$. Since one intercept is at $B(2)$, the other intercept $C$ must be sitting symmetrically at $5 + 3 = 8$ units!

Answer 1

The Correct Option is B

Concept: Translating a parabola changes the coordinates of its vertex while keeping its structural shape and width parameters perfectly identical. If the vertex moves to a new position $(h, k)$, the equation of the translated parabola becomes $y - k = -(x - h)^2$. Step 1: Find the vertex and intercepts of the original parabola.
The given equation of the initial parabola is $y = 4 - x^2$.
• The vertex $P$ occurs at the maximum point where $x = 0$, giving $y = 4$. Thus, $P = (0, 4)$.
• To find the $x$-intercepts $A$ and $B$, set $y = 0$: \[ 4 - x^2 = 0 \quad \Rightarrow \quad x = \pm 2 \] This gives the intercept points $A(-2, 0)$ and $B(2, 0)$.

Step 2: Set up the equation for the translated parabola.
The problem states that the vertex is shifted along the line $y = x + 4$. Let the coordinates of the new translated vertex be $P'(h, k)$. Since it lies on this line, its coordinates satisfy: \[ k = h + 4 \] Using this new vertex, write the equation of the translated parabola: \[ y - k = -(x - h)^2 \quad \Rightarrow \quad y - (h + 4) = -(x - h)^2 \quad \cdots (1) \]

Step 3: Apply the intercept condition to find the vertex coordinates.
We are given that the new translated parabola must pass through the original intercept point $B(2, 0)$. Substitute $x = 2$ and $y = 0$ into equation (1): \[ 0 - (h + 4) = -(2 - h)^2 \quad \Rightarrow \quad h + 4 = (2 - h)^2 \] Expand the quadratic expression on the right side: \[ h + 4 = 4 - 4h + h^2 \] Cancel out the constant 4 from both sides and collect all terms on one side: \[ h^2 - 5h = 0 \quad \Rightarrow \quad h(h - 5) = 0 \] This gives two possible values for the horizontal shift: $h = 0$ (the original unshifted position) or the true translated shift: \[ h = 5 \]

Step 4: Find the new intercept point $C$.
Substitute $h = 5$ back into equation (1) to get the final explicit equation of the translated parabola: \[ y - (5 + 4) = -(x - 5)^2 \quad \Rightarrow \quad y - 9 = -(x - 5)^2 \] To find the new $x$-intercept points $B$ and $C$, set $y = 0$: \[ 0 - 9 = -(x - 5)^2 \quad \Rightarrow \quad 9 = (x - 5)^2 \] Take the square root on both sides: \[ x - 5 = \pm 3 \] This yields our two intercept solutions:
• $x_1 = 5 - 3 = 2$ (This is the shared intercept point $B$, confirming our algebra)
• $x_2 = 5 + 3 = 8$ (This is the new intercept point $C$) Therefore, the abscissa ($x$-coordinate) of point $C$ is exactly 8, matching option (B).