Problem 58 - A Graphical Approach to College Algebra

Question

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A circle and a parabola; one point.

Step-by-Step Solution

Answer

Draw a circle \(x^2 + y^2 = 1\) and a parabola \(y = x^2\), intersecting tangentially at \((0,1)\).

Step 1: Understand the Problem

We need to draw a circle and a parabola that intersect at precisely one point. This means the two curves should touch each other tangentially at one location.

Step 2: Sketch the Circle

Let's sketch the circle with a center at the origin (0,0) and a certain radius. Mathematical representation of a circle is usually \( x^2 + y^2 = r^2 \). For simplicity, choose \( x^2 + y^2 = 1 \).

Step 3: Sketch the Parabola

Next, we need to sketch a parabola. A simple way is to consider the equation \( y = x^2 \). This parabola opens upwards and its vertex is at the origin (0,0).

Step 4: Position the Circle and Parabola for Tangency

The circle with equation \( x^2 + y^2 = 1 \) and parabola \( y = x^2 \) will be tangent at the point \( (0,1) \). To ensure they only touch at one point, we check that substituting \( y = x^2 \) into the circle's equation results in the same point.

Step 5: Confirm Intersection Condition

Substitute \( y = x^2 \) into \( x^2 + y^2 = 1 \) and solve: \( x^2 + (x^2)^2 = 1 \Rightarrow x^2 + x^4 = 1 \). This equation simplifies to \( x^4 + x^2 - 1 = 0 \). The solution for this equation should have exactly one solution, which is verified as \( x = 0 \) giving the point \( (0,1) \).

Step 6: Finalize Sketch

Draw the circle and parabola ensuring they touch at only (0,1). Ensure that there is only tangency and no additional intersection.

Key Concepts

Circle Equation

The circle is a fundamental geometric shape often represented in mathematics by its equation. The equation for a circle centered at the origin

• is given by
  \[ x^2 + y^2 = r^2 \]
  where \( r \) is the radius of the circle.

The beauty of this equation lies in its simplicity, as it encompasses all the points that are equidistant from the center located at

• the origin (0,0). The term \( r^2 \) represents the radius squared.
• So, if \( r = 1 \), then the equation becomes \( x^2 + y^2 = 1 \), indicating a unit circle.

To sketch this, simply plot all points at a distance of 1 from the origin in all directions.
This creates a circle with a radius of 1 unit around the origin.

Parabola Equation

A parabola is a curve that plays a significant role in mathematics. It can be represented by its equation,

• commonly written in the form:
  \[ y = ax^2 + bx + c \]
  where \( a, b, \) and \( c \) are constants.

For simplicity, we'll consider a basic form:

• \( y = x^2 \), which represents a parabola with its vertex at the origin (0,0).
• This way, the parabola opens upward.
  The shape is symmetrical about the y-axis.

The further away you move from the vertex (both left and right), the higher the values of y due to the squared term increasing.
Sketching this parabola involves plotting points that solve this equation, clearly showing its upward-opening nature.

Intersection Points

Intersection points of two graphs are crucial as they indicate where the graphs meet or cross each other. In our context, we need both the circle and the parabola to have precisely one intersection point.

• This intersection occurs where both equations are satisfied simultaneously.
  In mathematical terms, it means solving the two equations together.

In our case, substituting the parabola equation \( y = x^2 \) into the circle's equation gives:

• \[ x^2 + (x^2)^2 = 1 \]
• This simplifies to \( x^4 + x^2 - 1 = 0 \).
• We want exactly one real solution, ensuring a single intersection point.

By analyzing, you'll find that \( x = 0 \) results in \( y = 1 \), indicating the intersection point at (0,1).
This is where the touch or tangency occurs.

Tangency

Tangency between two curves is a fascinating geometric concept. It refers to the situation where two curves touch at exactly one point without crossing each other.
In the context of our circle and parabola, tangency is achieved when these two shapes meet at only one point at (0,1).To ensure tangency in a mathematical sense, we

• substitute the equation of one curve into the other.
• Here, substituting \( y = x^2 \) into \( x^2 + y^2 = 1 \) verifies that only (0,1) satisfies both equations together.

This means the point of tangency is confirmed. It's not just an intersection; it's more about the exact touching at one point.
It is like gentle contact or a kiss of the curves rather than crossing or overlapping. Such an approach ensures the clarity in their interaction at that single point.