Question

write an equation of the parabola with vertex (4, 1) and focus (-5, 1). use the variables x and y. an equation of the parabola is

Explanation:

Step1: Determine the direction and value of \(p\)

Since the \(y -\)coordinates of the vertex \((4,1)\) and focus \((- 5,1)\) are the same, the parabola opens horizontally.
The formula for \(p\) (distance from vertex to focus) is \(p=x_{focus}-x_{vertex}\).
So, \(p=-5 - 4=-9\).

Step2: Use the standard form of a horizontal parabola

The standard form of a horizontal parabola is \((y - k)^{2}=4p(x - h)\), where \((h,k)\) is the vertex.
Here, \(h = 4,k = 1,p=-9\).
Substitute these values into the formula: \((y - 1)^{2}=4\times(-9)(x - 4)\).

Step3: Simplify the equation

\((y - 1)^{2}=-36(x - 4)\)

Answer:

\((y - 1)^{2}=-36(x - 4)\)