Problem 1 - A Graphical Approach to Precalculus with Limits

Question

Fill in the blank to correctly complete each sentence. For the plane curve defined by $$ x=t^{2}+1, y=2 t+3, \quad \text { for } t \text { in }[-4,4] $$, the ordered pair that corresponds to \(t=-3\) is__________.

Step-by-Step Solution

Answer

The ordered pair is (10, -3).

Step 1: Understand the Problem

We need to find the ordered pair \(x, y\) for the given parametric equations \(x = t^2 + 1\) and \(y = 2t + 3\) when \(t = -3\). This means substituting \(t = -3\) into both equations to find the values of \(x\) and \(y\).

Step 2: Substitute into the x Equation

Substitute \(t = -3\) into the equation \((x = t^2 + 1)\). Calculate as follows: \(x = (-3)^2 + 1\) Simplify the expression: \(x = 9 + 1 = 10\).

Step 3: Substitute into the y Equation

Substitute \(t = -3\) into the equation \((y = 2t + 3)\). Calculate as follows: \(y = 2(-3) + 3\) Simplify the expression: \(y = -6 + 3 = -3\).

Step 4: Formulate the Ordered Pair

Using the results from Steps 2 and 3, the ordered pair \(x, y\) for \(t = -3\) is \((10, -3)\).

Key Concepts

Ordered Pair

An ordered pair is a fundamental concept in mathematics, particularly in coordinate geometry. It is typically represented as \( (x, y) \), where \( x \) and \( y \) are the coordinates on the Cartesian plane. This pair gives specific information about a point's location by giving its horizontal (\( x \)) and vertical (\( y \)) positions.The term "ordered" is crucial because the sequence matters. In \( (x, y) \), \( x \) always comes before \( y \). Swapping them, to make \( (y, x) \), would represent a different position on the plane, unless \( x = y \).In the context of parametric equations, an ordered pair results from evaluating the equations at a specific parameter. For example, in the given exercise, the equations for \( x = t^2 + 1 \) and \( y = 2t + 3 \) help us find a point by substituting a particular \( t \) value. By doing this, we get an ordered pair that precisely marks a spot on the curve defined by these equations.

Substitute

Substitution is a critical step in solving equations, especially in parametric forms. It involves replacing a variable with a specific numerical value to perform calculations and find a solution. In the exercise, we use the parametric equations: \( x = t^2 + 1 \) and \( y = 2t + 3 \). The task is to find the values of \( x \) and \( y \) for \( t = -3 \).By substituting, we replace every occurrence of \( t \) in both equations with \( -3 \). This leads to two simpler equations that we can solve immediately:

• \( x = (-3)^2 + 1 \)
• \( y = 2(-3) + 3 \)

Through substitution, abstract parametric equations become concrete values, paving the way for further simplifications and solutions.

Simplify Expression

Simplifying an expression means reducing it to its most straightforward form. It often involves arithmetic operations like addition, subtraction, multiplication, and occasionally division. Simplified expressions are easier to interpret and utilize for problem-solving.In the exercise, after substituting the value of \( t = -3 \) into the equations:

• The x-equation becomes \( x = 9 + 1 \)
• The y-equation becomes \( y = -6 + 3 \)

The next step is simplification:

• For the \( x \) equation, combine the numbers to get \( x = 10 \)
• For the \( y \) equation, combine to find \( y = -3 \)

This simplification takes complex algebraic forms and turns them into straightforward number results, critical for identifying the ordered pair as \( (10, -3) \).