91Ó°ÊÓ

A circle equation is a mathematical way to describe all the points that are a fixed distance, known as the radius, from a central point called the center. In its standard form, the equation of a circle with a center at the origin \((h, k)\) and radius \(r\) is given by:

• \( (x - h)^2 + (y - k)^2 = r^2 \)

In the exercise example, the circle equation is \( x^2 + y^2 = 25 \). Here, the center is at the origin \((0,0)\) and the radius is 5, since \( \sqrt{25} = 5\).
This means every point \((x, y)\) on the circle is exactly 5 units away from the origin. To graph it, you start at the center point, \((0,0)\), and stretch out to a distance of 5 units in all directions, forming a circle.
Graphically, this creates a perfect round shape centered at the origin which is essential in solving graphical problems.

A parabola is a symmetrical open plane curve formed by points equidistant from a fixed point (focus) and a straight line (directrix). The standard form of a parabola equation is:

• \( y = ax^2 + bx + c \)

In this particular exercise, the parabola equation is derived from \(3x^2 - 16y = 0\), which we can rearrange to get \(y = \frac{3}{16}x^2\).
This equation indicates a parabola opening upwards, with its vertex at the origin \((0, 0)\). The factor \(\frac{3}{16}\) determines the "width" or "steepness" of the parabola. Graphing this involves plotting several \((x, y)\) points by choosing values for \(x\) and calculating \(y\), and then drawing a smooth curve through these points. The parabola will appear as a U-shaped curve centered vertically around the y-axis.

The intersection of graphs refers to the points where two different functions meet on a coordinate plane. In a system of equations like ours, finding these intersections graphically involves plotting both equations on the same set of axes.

• For the given exercise, it involves plotting the circle \(x^2 + y^2 = 25\) and the parabola \(y = \frac{3}{16}x^2\) on the same graph.

These intersection points represent solutions to the system, as they satisfy both equations simultaneously.
Finding these points visually can sometimes be imprecise, so confirming them by algebraically checking the coordinates against the original equations is crucial.Observing the graphs, these intersections occur at specific coordinates where both shapes overlap. Accurately identifying or calculating these points provides the solution to the system of equations. It's a process that combines both graphical and algebraic methods for verification.