Problem 55 - A Graphical Approach to Precalculus with Limits

Question

Draw a sketch of the two graphs described with the indicated number of points of intersection. A line and a circle; two points

Step-by-Step Solution

Answer

The line intersects the circle at two distinct points, confirming two intersections.

Step 1: Understand the Problem

We are asked to sketch a line intersecting a circle at two points. This means the line should pass through the circle in such a way that it enters and exits the circle, creating exactly two intersection points.

Step 2: Sketch the Circle

Draw a circle on a coordinate plane. A circle is defined by its center and radius. For simplicity, let's assume we have a circle centered at the origin with a radius of 3 units. This circle will have the equation \(x^2 + y^2 = 9\).

Step 3: Sketch the Line

Next, choose an equation for a line that will intersect the circle at two points. A candidate simple line is \(y = x + 1\), which typically will intersect the circle based on its position relative to the center of the circle.

Step 4: Determine Points of Intersection

To confirm the line intersects the circle at two points, substitute \(y = x + 1\) into the circle's equation: \(x^2 + (x+1)^2 = 9\). Simplify the equation to find the x-coordinates of the intersection points: \(x^2 + x^2 + 2x + 1 = 9\) leads to \(2x^2 + 2x - 8 = 0\). Solve \(x^2 + x - 4 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the x-values.

Step 5: Calculate Intersection Points

Apply the quadratic formula to \(x^2 + x - 4 = 0\), where \(a=1\), \(b=1\), and \(c=-4\). Calculate \(x = \frac{-1 \pm \sqrt{1^2 - 4 \ imes 1 \ imes (-4)}}{2 \ imes 1}\) or \(x = \frac{-1 \pm \sqrt{1 + 16}}{2}\) giving \(x = \frac{-1 \pm \sqrt{17}}{2}\). These are the x-values for the intersection.

Step 6: Verify Points of Intersection with the Line

With the x-values from the previous step, substitute back into \(y = x + 1\) to find y-values. This provides two intersection points explicitly, confirming the two points of intersection between line and circle exist.

Step 7: Finalize the Sketch

On your sketch, ensure the line \(y = x + 1\) clearly crosses the circle at the computed points. There, the line appears to enter and exit the circle, representing the two intersection points visually.

Key Concepts

Intersection Points

When discussing intersection points in the context of graph sketching, we're talking about specific locations on a coordinate plane where two graphs meet. In this case, the graphs are a line and a circle. An intersection point between these two graphs is a solution to both their equations simultaneously.

• The intersection means that both the x and y coordinates satisfy the equations of the line and the circle.
• If a line intersects a circle at two points, it enters the circle at one point and exits at another.
• This is visually represented in our example by the line entering and exiting the circle boundary.

The process involves finding exact coordinates by solving the equations together, either algebraically by substitution or using graphical estimations through sketching. Whether drawing or calculating, these points show the shared positions between two different types of graphs.

Circle Equation

A circle's equation primarily defines its geometric properties on a coordinate plane, depicting all the points at an equal distance from a single central point—its center. The standard form of a circle's equation is \[x^2 + y^2 = r^2\] where:

• \(x^2 + y^2 = r^2\) - represents every point's distance from the center being equal to the radius \(r\).
• Center: The origin by default if the equation represents \(x^2 + y^2 = r^2\), assuming no translations.
• Radius: In our example, it is 3, since \(r^2\) is 9, making \(r\) the square root of 9.
• Every point \((x, y)\) on the circle maintains the equation's correctness, illustrating a constant radius.

This understanding helps in plotting and verifying points accurately on a graph to sketch an exact circle.

Line Equation

Line equations are critical in defining linear relationships between variables in a coordinate plane. The standard form used here is \[y = mx + c\], which denotes:

• Slope \(m\): The measure of steepness or gradient of the line; it shows the rate of change of \(y\) with respect to \(x\).
• Intercept \(c\): The point where the line crosses the y-axis; it reflects the value of \(y\) when \(x = 0\).

For our line, \(y = x + 1\), the slope is 1, signifying a 45-degree angle line upward to the right, and the y-intercept is 1, displaying a starting point one unit above the origin on the y-axis. This equation selection is strategic to guarantee intersection points with the circle, ensuring the line both enters and exits through the circle's boundary, hence fulfilling the exercise's requirements.

Coordinate Plane

The coordinate plane is an essential framework used in graphing and interpreting data with two dimensions: the x-axis (horizontal) and y-axis (vertical). It enables visualization and solves algebraic and geometric problems involving shapes, lines, and other figures.

• This plane, also known as the Cartesian plane, is divided into four quadrants, helping to locate points using (x, y) coordinates.
• Our focus remains in calculating and interpreting these coordinates when graphing equations like lines and circles.
• The origin, (0, 0), is the intersection of the axes and is often the starting point for plotting.

Understanding the coordinate plane ensures precise placements and dimensions of graphs, facilitating accurate sketches and solutions, such as determining where a line intersects a circle at specific points.