Problem 25 - College Algebra

Question

Find the standard form of the equation of each parabola satisfying the given conditions. Vertex: \((2,-3) ;\) Focus: \((2,-5)\)

Step-by-Step Solution

Answer

The standard form of the equation for the parabola is: \(y=(1/8)*(x-2)^2 - 3\) or upon simplification \(y= (1/8)x^2 - (1/2)x + 1/2\)

Step 1: Identify values from the problem

From the problem, the vertex \((h,k) = (2,-3)\) and the focus \((h, f_{y}) = (2,-5)\). Here, notice that the value of x-coordinate is same for both vertex and focus which signifies that our parabola either opens upwards or downwards.

Step 2: Identify the orientation of the parabola and calculate the value of a

As the y-values of the focus is lesser than the y-values of the vertex, the parabola opens downwards. Calculate the value of a using the formula \(a=1/(4f)\). Here, f is the distance from the vertex (h,k) to the focus (h,f_{y}). Therefore, f = k - f_{y}, i.e., -3 - (-5)=2. Substituting f=2 in \(a=1/(4f)\) gives a = 1/8.

Step 3: Substitute values in standard parabolic equation

The standard form equation of the parabola should be \(y=a(x-h)^2+k\) as it opens upwards or downwards. Substitute the known values (h,k)=(2,-3) and a=1/8 in the equation.

Step 4: Simplify the equation

Substitute the identified values into the standard form equation to get \(y=(1/8)*(x-2)^2 - 3\). Expand and simplify this equation to obtain the final result in standard form.