Question
What are the vertex, focus, and directrix of the parabola whose equation is \( y-4=\frac{1}{4}(x-7)^{2} \) ? vertex \( =(\square) \) focus \( =(\square \)
Ask by Robbins Wheeler. in the United States
Feb 20,2025
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Tutor-Verified Answer
Answer
Vertex = (7, 4), Focus = (7, 5), Directrix = y = 3
Solution
The given equation of the parabola is
y – 4 = (1/4)(x – 7)².
This equation is in the vertex form of a parabola that opens upward or downward:
y – k = (1/(4p))·(x – h)²,
where (h, k) is the vertex and p is the distance from the vertex to the focus.
From the equation:
1/(4p) = 1/4 ⟹ p = 1,
and the vertex (h, k) is:
(7, 4).
Since the parabola opens upward (because the coefficient 1/4 is positive), the focus lies p units above the vertex. Therefore:
Focus = (7, 4 + 1) = (7, 5).
The directrix is the horizontal line p units below the vertex:
Directrix: y = 4 – 1 = 3.
So, the answers are:
Vertex = (7, 4)
Focus = (7, 5)
Directrix = y = 3.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
The equation you provided can be rewritten to identify the vertex, focus, and directrix. This equation is in the form \( y - k = \frac{1}{4p}(x - h)^2 \), where \((h, k)\) is the vertex. In your case, the vertex is at \((7, 4)\). To find the focus and directrix, we need to determine \( p \). Since the coefficient \(\frac{1}{4}\) corresponds to \(\frac{1}{4p}\), we can equate \(4p = 4\), giving us \(p = 1\). The focus is at \((h, k + p) = (7, 4 + 1) = (7, 5)\), and the directrix is determined by the formula \(y = k - p\), which gives \(y = 4 - 1 = 3\). So, vertex \( =(7, 4) \), focus \( =(7, 5) \), and directrix \( = y = 3\).
