Problem 8 - Precalculus

Question

Graph the parametric equations after eliminating the parameter t. Specify the direction on the curve corresponding to increasing values of \(t\). \(t\) can be any real number. $$x=t+1, y=t^{2}$$

Step-by-Step Solution

Answer

The graph is a parabola opening upwards, moving left to right as \(t\) increases, with the equation \(y = (x - 1)^2\).

Step 1: Understand the Given Equations

We have two parametric equations: \(x = t + 1\) and \(y = t^2\). These equations describe points on a graph where \(x\) and \(y\) are dependent on the parameter \(t\). The goal is to combine these into one single equation describing the curve without the parameter \(t\).

Step 2: Solve for Parameter t

From the equation \(x = t + 1\), solve for \(t\):\[ t = x - 1 \].

Step 3: Substitution into y Equation

Substitute \(t = x - 1\) into the second equation \(y = t^2\):\[ y = (x - 1)^2 \].

Step 4: Simplification and Description

The new equation is \(y = (x - 1)^2\). This is the equation of a parabola, opening upwards, with its vertex at the point \((1, 0)\).

Step 5: Determine Direction on the Curve

To find the direction as \(t\) increases, observe that starting from \(t = -\infty\), the parabola moves from left to right. This means as \(t\) increases, the curve moves from left to right along the parabola with increasing \(x\) values.

Key Concepts

Eliminating the Parameter

Eliminating the parameter in parametric equations refers to rewriting the system of equations so that one equation expresses the relationship between the two variables directly, without involving the parameter.
This makes it simpler to understand and visualize the geometric shape represented by the equations.

• Start with identifying the parametric equations: in this case, we have \( x = t + 1 \) and \( y = t^2 \).
• Find the parameter \( t \) from one of the equations. Solving for \( t \) from \( x = t + 1 \), we get \( t = x - 1 \).
• Substitute this expression for \( t \) into the second equation, \( y = t^2 \), resulting in \( y = (x - 1)^2 \).

By eliminating the parameter \( t \), we've transformed the system into a single equation \( y = (x - 1)^2 \), revealing the underlying relationship between \( x \) and \( y \). This is particularly useful for graphing.

Graphing Parabolas

When dealing with equations like \( y = (x - 1)^2 \), which result from eliminating the parameter, it's essential to recognize their structure.
The equation \( y = (x - 1)^2 \) is a standard form of a parabola.

• The vertex form \( y = a(x-h)^2 + k \) identifies the vertex at \((h, k)\). Here, the equation is \( y = (x - 1)^2 \), indicating a vertex at \((1, 0)\).
• The parabola opens upwards as the coefficient of \( (x-1)^2 \) is positive.
• To graph, start at the vertex \((1,0)\) and draw the curve symmetrically about the axis through the vertex.

Understanding these characteristics facilitates graphing since parabolas have a distinctive U-shape and predictable behavior.

Direction of the Curve

Understanding the direction of a curve is essential when analyzing parametric equations, especially as the parameter influences how the curve is traced.
For the given equations \( x = t + 1 \) and \( y = t^2 \):

• As \( t \) increases, notice how both \( x \) and \( y \) change. Since \( x = t + 1 \), it linearly increases with \( t \).
• The curve starts from the left, moving to the right as \( t \) moves from negative infinity to positive infinity. Thus, the path is traced from left to right.
• In this setup, as \( t \) becomes more positive, \( y = t^2 \) increases faster than \( x \), causing the steep upward slant typical of a parabola.

Graphing with parametric equations provides rich insight into the motion and behavior of points along the curve.