Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Alternative Parametric Forms Give two parametric representations of the parabola \(y=a(x-h)^{2}+k\)

Step-by-Step Solution

Answer

Parametrically: 1) \( x = t, y = a(t-h)^2+k \); 2) \( x = h \pm \sqrt{\frac{y-k}{a}}, y = t \).

Step 1: Understanding the Parabola Equation

The given equation of a parabola is in vertex form: \( y = a(x-h)^2 + k \), where \( h \) and \( k \) are the coordinates of the vertex, and \( a \) determines the direction and width of the parabola.

Step 2: Choose a Parameter

To represent the parabola in parametric form, we introduce a parameter, typically \( t \), to replace one of the variables. Common choices involve letting \( x \) equal \( t \).

Step 3: Express \( x \) as a Parameter

We choose \( x = t \) as the parameter. Thus, \( x = t \) is our first parametric equation.

Step 4: Substitute for \( y \) in Terms of \( t \)

Using \( x = t \), substitute into the parabola equation: \( y = a(t-h)^2 + k \). This is our second parametric equation.

Step 5: Alternative Choice of Parameter

For an alternative parametric form, set \( y = t \) and rearrange the equation \( t = a(x-h)^2 + k \) to solve for \( x \) in terms of \( t \).

Step 6: Solve for \( x \)

Rearrange \( t = a(x-h)^2 + k \) to get \((x-h)^2 = \frac{t-k}{a} \). Solving for \( x \) gives us \( x = h \pm \sqrt{\frac{t-k}{a}} \).

Step 7: Finalize the Alternative Parametric Equations

The alternative parametric equations can be \( y = t \) and either \( x = h + \sqrt{\frac{t-k}{a}} \) or \( x = h - \sqrt{\frac{t-k}{a}} \), depending on the root chosen.

Key Concepts

Parabola

A parabola is a U-shaped curve that is common in mathematics, representing quadratic functions graphically. Imagine drawing a smile or a frown; that's basically a parabola. These curves are symmetrical, meaning one side mirrors the other. In the context of the equation, the standard form for a parabola is either upward or downward based on the coefficient of the squared term.

• If the coefficient is positive, the parabola opens upwards, like a smile.
• If it's negative, the parabola opens downwards, resembling a frown.

Parabolas have unique properties, such as a directrix line and a focus point, which define its precise shape. A key feature is the vertex, which is the turning point of the curve. Understanding parabolas is essential for grasping more complex topics in math, such as optimization problems or projectile motion.

Vertex Form

The vertex form of a quadratic equation is written as \( y = a(x-h)^2 + k \). This form is useful because it provides the vertex of the parabola directly, which is \((h, k)\).

• \( h \)
• k

These represent the x-coordinate and y-coordinate of the vertex, respectively. The parameter \( a \) affects how wide or narrow the parabola looks and whether it faces up or down.

• If \( |a| > 1 \), the parabola is narrow.
• If \( |a| < 1 \), the parabola is wide.

The vertex form simplifies many analyses in math, especially when optimizing a function or determining minimal or maximal values.

Parameterization

Parameterization is the process of expressing a mathematical expression using parameters. In our scenario, we want to represent a parabola using parameters other than \( x \) and \( y \). Typically, we choose a parameter \( t \) and express \( x \) and \( y \) in terms of \( t \).
This approach is valuable because it allows us to describe the curve's behavior in a flexible way, often simplifying calculations or assisting in rendering graphics.

• For a parabola, you can set \( x = t \) and then solve for \( y \), resulting in \( y = a(t-h)^2 + k \).
• Alternatively, setting \( y = t \) and solving for \( x \), provides another set of parametric equations.

Parametric equations are particularly useful in physics and engineering for modelling trajectories and motions.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, usually presented as \( f(x) = ax^2 + bx + c \). These functions always produce a parabolic graph. The graph's orientation and shape depend on the values of \( a \), \( b \), and \( c \).
The general properties of quadratic functions include:

• The vertex, which is the highest or lowest point.
• The axis of symmetry, a vertical line that divides the parabola into mirror images.
• The direction of opening, determined by the sign of coefficient \( a \).

Understanding quadratic functions is crucial because they model various real-world scenarios, from physics to finance. They help solve problems where you need to find minimum or maximum values, such as maximizing profit or minimizing cost.