Problem 5 - Precalculus with Calculus Previews

Question

Sketch the curve that has the given set of parametric equations. $$ x=\sqrt{t}, y=5-t, \quad t \geq 0 $$

Step-by-Step Solution

Answer

The curve is a downward-opening parabola with vertex at (0, 5), starting from x=0.

Step 1: Understanding the Parametric Equations

We are given that the parametric equations are \( x = \sqrt{t} \) and \( y = 5 - t \) with \( t \geq 0 \). This means that the parameter \( t \) represents time, and as \( t \) increases, \( x \) and \( y \) change accordingly.

Step 2: Eliminate the Parameter

To sketch the curve, we commonly eliminate the parameter \( t \). We start with the equation for \( x \): \( x = \sqrt{t} \). Squaring both sides, we get \( t = x^2 \). Substitute this expression for \( t \) into the equation for \( y \): \( y = 5 - x^2 \).

Step 3: Identify the Curve

The resulting equation \( y = 5 - x^2 \) is a quadratic equation representing a parabola that opens downwards. The vertex of this parabola is at the point \( (0, 5) \).

Step 4: Determine Domain and Range

Since \( t \geq 0 \), and \( x = \sqrt{t} \), \( x \) is always non-negative (\( x \geq 0 \)). The range of \( y = 5 - x^2 \) is determined by the maximum value of \( y \), which is 5 at \( x = 0 \). Therefore, \( y \) decreases as \( x \) increases.

Step 5: Sketch the Curve

On the coordinate plane, plot the point where the curve starts, at \( (0, 5) \). As \( x \) increases from 0, \( y \) will decrease following the parabola \( y = 5 - x^2 \). The curve is in the first and second quadrants, extending downward from the vertex at \( (0, 5) \).

Key Concepts

Curve Sketching

Curve sketching is a graphical technique used to draw the curve based on given equations. With parametric equations, like in this exercise, each coordinate \((x, y)\) on the plane is defined by a separate function of a parameter \( t \). These parametric functions describe how the \( x \) and \( y \) coordinates change as \( t \) varies.

In the given example, where \( x = \sqrt{t} \) and \( y = 5 - t \), we start by eliminating the parameter. Rewriting the equations helps in determining a clear relation between \( x \) and \( y \). Once we eliminate the parameter, the problem becomes simpler, turning into a problem of recognizing known types of equations, like a quadratic equation.

Here are some tips for effective curve sketching with parametric equations:

• Identify how each parameter \( t \) affects \( x \) and \( y \). You are essentially watching how the curve evolves as \( t \) changes.
• Think about the geometry: As \( t \) increases, consider how the shape and direction of the curve develop.
• Once the parameter is removed, see if the resulting equation reminds you of any common graphs.

By understanding these components, curve sketching becomes an intuitive process based on recognizing patterns and leveraging known curve shapes.

Quadratic Equation

A quadratic equation is a second-degree polynomial, typically described by the form \( y = ax^2 + bx + c \). In many cases, quadratic equations describe parabolas, which is also the result in this exercise.

The parabola from this problem is described by the equation \( y = 5 - x^2 \). This is a transformation of the basic form \( y = x^2 \), but inverted due to the negative sign. Each quadratic equation unfolds into a specific shape, with its opening determined by the sign of \( a \) (negative here, resulting in a downward-opening parabola).

Key properties to remember with quadratic equations:

• The coefficient \( a \) (positive or negative) dictates whether the parabola opens up or down.
• The vertex form of the equation \( y = a(x-h)^2 + k \) reveals the vertex \( (h, k) \), which is the highest or lowest point on the curve.
• In \( y = 5 - x^2 \), the vertex is \( (0, 5) \), showcasing the maximum point of the parabola.

Understanding these aspects of quadratic equations allows students to visualize the curve quickly and accurately without relying solely on plotting individual points.

Parabola

A parabola is a symmetrical, U-shaped curve that can open upwards or downwards on a graph. Parabolas are the graphical representation of quadratic equations.

In this exercise, after transforming the parametric equations to \( y = 5 - x^2 \), we realize the curve is a downward-opening parabola with its vertex at \( (0, 5) \). This is because of the negative sign in front of \( x^2 \), and the vertex represents the maximum point in the curve.

Some important features of parabolas include:

• The direction (up or down) is determined by whether the \( x^2 \) term is positive or negative.
• The axis of symmetry is a vertical line through the vertex. In this problem, the axis of symmetry is the \( y \)-axis itself.
• A parabola continues indefinitely in its opening direction, broadening the further from the vertex it gets.

This understanding of parabolas helps simplify predicting the behavior of quadratic relationships, making them less daunting and more manageable. Being familiar with these properties can lead to more confidence in handling related math problems.