Problem 91 - A Graphical Approach to Precalculus with Limits

Question

Find an equation of a parabola that satisfies the given conditions. Horizontal axis; vertex \((-2,3)\); passing through \((-4,0)\)

Step-by-Step Solution

Answer

\( y = -\frac{3}{4}(x + 2)^2 + 3 \) is the equation of the parabola with the given conditions.

Step 1: Identify the General Equation of a Parabola with Horizontal Axis

The general equation for a parabola with a horizontal axis is \[ y = a(x - h)^2 + k, \]where \((h, k)\) is the vertex of the parabola. Given the vertex \((-2, 3)\), we can substitute it into the equation to get\[ y = a(x + 2)^2 + 3. \]

Step 2: Substitute the Given Point to Find 'a'

Substitute the point \((-4, 0)\) into the equation from Step 1 to find the value of \(a\).Set \( y = 0 \) and \( x = -4 \):\[ 0 = a(-4 + 2)^2 + 3. \]Simplify and solve:\[ 0 = a(-2)^2 + 3 \]\[ 0 = 4a + 3 \]\[ 4a = -3 \]\[ a = -\frac{3}{4}. \]

Step 3: Write the Equation of the Parabola

Substitute \(a = -\frac{3}{4}\) back into the equation derived in Step 1:\[ y = -\frac{3}{4}(x + 2)^2 + 3. \]This is the equation of the parabola that satisfies all the given conditions.

Key Concepts

Vertex Form

The vertex form of a parabola is a way to express the equation of a quadratic in a manner that easily reveals the coordinates of the vertex, which is the highest or lowest point on the parabola. The vertex form is given by \[ y = a(x - h)^2 + k \]where

• \( a \) controls the direction and the width of the parabola.
• \( (h, k) \) is the vertex of the parabola.

Using this form, it's straightforward to see the parabola's vertex in our problem: \( (h, k) = (-2, 3) \). This means the vertex is at \( x = -2, y = 3 \). Transforming the vertex form into an equation is simply a matter of plugging in the vertex coordinates. If you have points that the parabola passes through, you can adjust the parameter \( a \) to fit them.

Horizontal Axis

When dealing with parabolas, the axis of symmetry is the line that runs down the "center" of the parabola, dividing it into two mirror-image halves. Typically, in a standard "up or down" facing parabola, this would be a vertical line. However, with a parabola that has a horizontal axis, things are a bit different.
For a parabola with a horizontal axis of symmetry, the general form is slightly twisted:

• The equation becomes \[ y = a(x - h)^2 + k \].
• The vertex form is still applicable, but the axis spreads out horizontally rather than vertically.

In this form, the parabola opens to the left or right because it's the \( x \) term that becomes squared,meaning the symmetry shifts from what you typically expect in a vertical axis setting.

Substitution Method

The substitution method is a crucial tool when it comes to finding the missing constants in equations. In the case of our parabola problem, we use the substitution method to find the value of \( a \) in the parabola's equation. Imagine if you have a point \((-4, 0)\), through which your parabola passes.
Steps to apply the substitution method:

• Start with your vertex form equation: \[ y = a(x + 2)^2 + 3 \].
• Substitute the coordinates of the known point, \( (x, y) = (-4, 0) \), into the equation.

This means setting \( y = 0 \) and \( x = -4 \), yielding \[ 0 = a(-2)^2 + 3 \]. Solving this will give the value of \( a \), which is essential to have the complete equation of the parabola.

Solving Quadratics

Solving quadratic equations involves finding the values of the unknowns that satisfy the equation. In the parabola exercise, we ultimately need to solve for the term \( a \) to finalize the equation. Here is how solving the quadratic part unfolds in this specific problem:

• We have \[ 0 = 4a + 3 \].
• Rearranging the equation, we get \[ 4a = -3 \].
• Solving for \( a \), we find \[ a = -\frac{3}{4} \].

This value of \( a \) is substituted back into the vertex form to provide the final formulation of the parabola, ensuring all conditions, including the vertex and the additional point, are satisfied. Solving these equations involves basic algebra but requires attention to detail to avoid mistakes.