Problem 38 - A Graphical Approach to College Algebra

Question

Give two parametric representations for each plane curve. Use your calculator to verify your results. $$y=(x+1)^{2}+1$$

Step-by-Step Solution

Answer

The parametric representations are \( (t, (t+1)^2+1) \) and \( (t-1, t^2+1) \).

Step 1: Identifying the Given Cartesian Equation

The given Cartesian equation is a parabola: \[ y = (x+1)^2 + 1 \] This is in the form of \( y = a(x-h)^2 + k \) where \( a = 1 \), \( h = -1 \), and \( k = 1 \). This means the parabola is shifted 1 unit left and 1 unit up.

Step 2: First Parametric Representation

For the first parametric representation, let \( t = x \). Then, the parametric equations become:\[\begin{align*}x(t) &= t, \y(t) &= (t+1)^2 + 1\end{align*}\] This representation is straightforward where \( t \) takes on values corresponding to \( x \).

Step 3: Second Parametric Representation

For a second representation, consider a change of variable such as \( t = x + 1 \). This yields:\[\begin{align*}x(t) &= t - 1, \y(t) &= t^2 + 1\end{align*}\] Here, \( t \) represents the horizontal shift by 1 unit.

Step 4: Calculator Verification

Use a graphing calculator to plot both sets of parametric equations. Check that both parametric curves match the graph of the original Cartesian equation \( y = (x+1)^2 + 1 \). If plotted correctly, both parametric representations should overlap with the original curve.

Key Concepts

Cartesian Equation

A Cartesian equation is a mathematical expression involving variables, often denoted as \( x \) and \( y \), that defines the relationship between them. In this case, the Cartesian equation of interest is the parabola given by \( y = (x+1)^2 + 1 \).
This equation is written in the vertex form of a parabola, \( y = a(x-h)^2 + k \). Here, \( a = 1 \), \( h = -1 \), and \( k = 1 \).

• \( a \) determines the direction and width of the parabola.
• \( h \) represents the horizontal shift from the origin.
• \( k \) indicates the vertical shift.

Understanding that this parabola is a vertically oriented one, shifted one unit left and one unit up, helps in visualizing its graph. Plotting this equation will reveal a U-shaped curve positioned around the vertex \((-1,1)\). As you work with parametric equations, expressing the same curve in terms of parameters rather than direct \( x \) and \( y \) variables is key.

Plane Curve

A plane curve is a curve that lies flat on a two-dimensional surface such as paper or a graph plane. Essentially, it represents a path traced by a moving point that has a given or ascertainable direction at each position.
In the context of our original exercise, the plane curve represented by the equation \( y = (x+1)^2 + 1 \) is a parabola.
When working with such curves, it is common to explore them in parametric forms to provide alternative ways to describe the same curve and facilitate different types of analysis.

• The first parametric representation uses \( t = x \), thereby providing equations \( x(t) = t \) and \( y(t) = (t+1)^2 + 1 \).
• The second representation, using \( t = x + 1 \), forms equations \( x(t) = t-1 \) and \( y(t) = t^2 + 1 \).

These representations are useful in both graphical and analytical explorations, offering insights into the nature and properties of the plane curve.

Parabola

A parabola is a specific type of plane curve often associated with quadratic functions. Its defining equation in vertex form is \( y = a(x-h)^2 + k \), making it easy to identify the vertex's position and the parabola's orientation.
In mathematics, a parabola has several distinctive features:

• Symmetrical around its vertical axis.
• The direction of its opening is determined by the sign of \( a \) (positive for upward, negative for downward).
• The vertex \( (h, k) \) is either a point of maximum or minimum value, depending on the parabola's orientation.

For the given parabola \( y = (x+1)^2 + 1 \), these characteristics include:

• Vertex at \((-1, 1)\).
• Opens upwards due to \( a = 1 \).

Translation into parametric representations helps demonstrate these properties through a different lens while adding flexibility in the way the curve can be handled mathematically or graphically. Parametric equations become particularly helpful for integrating the curve into broader mathematical contexts where time or another independent variable governs changes in \( x \) and \( y \).