Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex: \((2,-1) ;\) point: \((4,-3)\)

Step-by-Step Solution

Answer

The equation of the parabola is \( y=-0.5(x-2)^2-1 \).

Step 1: Setup the parabola equation

Start with the standard form of a parabola equation \( y=a(x-h)^2+k \) given \( h=2 \) and \( k=-1 \), this becomes \( y=a(x-2)^2-1 \).

Step 2: Substitute the coordinates of the given point

Substitute \( x=4, y=-3 \) into the equation which leads to: \[ -3 = a(4-2)^2-1 \] . Simplifying the expression gives: \[ -3=a(2)^2-1 \] .

Step 3: Solve for 'a'

Solving this equation for \( a \) yields: \[ -3=a*4-1 \Rightarrow a=(-3+1)/4=-0.5 \] .

Step 4: Write the final equation of the parabola

Substitute back the value of \( a \) into the parabola's standard equation, giving the final equation: \[ y=-0.5(x-2)^2-1 \].

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a key feature that helps define its shape and position on the coordinate plane. Think of it as the "turning point" of the parabola.
For a parabola that opens upwards or downwards, the vertex is at its highest or lowest point.

• The vertex is given as a pair of coordinates, typically denoted as \((h, k)\).
• In the context of the vertex form equation \(y = a(x-h)^2 + k\), the coordinates \((h, k)\) represent the vertex's location.

In the given exercise, the vertex is \((2, -1)\), which means the parabola reaches its minimum value at this point.
This understanding lays the foundation for shaping the rest of the parabola.

Solving Equations

Solving equations involves finding the unknown variables that satisfy the equality.
In our case, we need to determine the value of \(a\) in the parabola's equation. This process is crucial because it defines the parabola's width and direction.

• Our setup starts with an equation derived from the parabola's standard form.
• We then substitute known values to isolate and solve for the unknown variable.

To solve for \(a\), substitute \(x = 4\) and \(y = -3\) into the equation, getting \[-3 = a(4-2)^2 - 1\].
Rearranging and simplifying gives us the value of \(a = -0.5\).
Hence, understanding solving equations is pivotal to finding the right shape for our parabola.

Coordinate Substitution

Coordinate substitution helps simplify equations using known values. This step is vital for verifying if a point lies on a curve.
Here’s how it works in the context of parabolas:

• Start with the parabola's equation in its standard form.
• Use the known point to substitute into the equation for \(x\) and \(y\).
• This approach allows determination of the variable \(a\).

For our exercise, substituting \(x = 4\) and \(y = -3\) provides us with an equation simple enough to solve manually.
Effectively using coordinate substitution reveals other characteristics of the parabola and verifies our final equation.

Standard Form of a Parabola

The standard form of a parabola is a specific way to write its equation that makes it easy to identify its key features.
The formula is written as \(y = a(x-h)^2 + k\):

• \(a\) changes the parabola's width and direction—if negative, the parabola opens downwards.
• \(h\) and \(k\) pinpoint the vertex's location on the graph.

Using the provided vertex \((2, -1)\), the equation becomes \(y = a(x-2)^2 - 1\), ready for substituting other known coordinates to solve for \(a\).
Understanding this form makes it straightforward to derive the parabola's equation and predict its graphical layout accurately.