Problem 54 - Precalculus: Functions and Graphs

Question

Graph the two equations on the same coordinate plane, and estimate the coordinates of the points of Intersection. $$9 x^{2}+y^{2}=9 ; \quad y=e^{x}$$

Step-by-Step Solution

Answer

The points of intersection are approximately (0, 1) and (0.51, 1.67).

Step 1: Rearrange the First Equation

The first equation is a circle given as \(9x^2 + y^2 = 9\). To graph it, solve for \(y\): \[y = \pm \sqrt{9 - 9x^2}\] This represents the upper and lower halves of the circle centered at the origin with a radius of 3.

Step 2: Analyze the Second Equation

The second equation \(y = e^x\) is an exponential function. The graph of \(e^x\) passes through \( (0,1) \) and continuously increases as \(x\) increases, never touching the x-axis, making it an increasing curve.

Step 3: Plot Both Graphs

Use graphing software or plot the equations manually. For the circle, graph the half-circles \(y = \sqrt{9 - 9x^2}\) and \(y = -\sqrt{9 - 9x^2}\). For the exponential graph, plot \(y = e^x\).

Step 4: Identify Points of Intersection

Observe both graphs on the coordinate plane. The exponential function intersects the circle at two points. Estimate these points by identifying where \(y = e^x\) crosses the half-circles from Step 1.

Step 5: Estimate Coordinates

Based on the graph, the points of intersection appear to be approximately \((0, 1)\) and \((0.51, 1.67)\). Ensure these intersections are precise by checking both \(y\) values satisfy both equations.

Key Concepts

Circle Equation

A circle equation in the form of \(9x^2 + y^2 = 9\) represents a circle on the coordinate plane. In this expression, the circle is centered at the origin \((0,0)\). The number 9 on the right side of the equation is the square of the radius of the circle. Thus, the radius is \(\sqrt{9} = 3\).

To graph the equation, it helps to solve for \(y\). The circle equation can be rewritten as:

• \(y = \sqrt{9 - 9x^2}\)
• \(y = -\sqrt{9 - 9x^2}\)

These two functions represent the upper and lower halves of the circle, respectively. When you plot these functions, they create a complete circle on the coordinate plane.

Exponential Function

An exponential function like \(y = e^x\) has its unique charm. In this particular function, \(e\) is the base of natural logarithms, approximately 2.718.

This function:

• Starts passing through the point \((0,1)\), meaning when \(x = 0\), \(y = e^0 = 1\).
• Always yields a positive value. As \(x\) increases, \(e^x\) grows rapidly.
• Is never zero for any real number \(x\). The graph gets closer to the x-axis as \(x\) becomes more negative but never actually meets it.

The exponential curve is always increasing and helps in modeling growth processes.

Intersection Points

Intersection points occur where two graphs meet on the coordinate plane. For the circle and exponential function, these points are where both equations have the same \((x, y)\) solutions.

From the solutions, the estimated intersection points were around \((0, 1)\) and \((0.51, 1.67)\). You find these points by plotting both equations:\(9x^2 + y^2 = 9\) for the circle and \(y = e^x\) for the exponential curve.

At these intersections, both the value from the circle equation and the exponential equation should nearly match. For practical and educational purposes, it’s essential to verify these intersections by plugging the point coordinates back into each equation.

Coordinate Plane

The coordinate plane is a two-dimensional space formed by two perpendicular axes, the x-axis, and y-axis. It is used to graph equations and visualize mathematical relationships.

On this plane:

• The horizontal axis is the x-axis, where values increase to the right and decrease to the left.
• The vertical axis is the y-axis, where values increase upward and decrease downward.
• The point where both axes meet is called the origin, represented by \((0,0)\).

Using the coordinate plane, you can graph the circle and exponential functions from our exercise. It helps in finding where these graphs intersect, as you overlay them on this common framework to look for where they share coordinates.