91Ó°ÊÓ

When it comes to graphing parabolas, understanding their properties is key. A parabola is a type of curve on a graph. Every parabola has some essential features such as a vertex, axis of symmetry, focus, and directrix.
The equation of a parabola can typically be expressed in the form \( y = ax^2 + bx + c \) when using Cartesian coordinates. The vertex of the parabola represents the point where the curve changes direction. In our problem, after eliminating the parameter, we found the equation \( y = (x-3)^2 - 9 \).
This equation shows a parabola that opens upwards. The vertex is at the point \(3, -9\). By analyzing the equation, we noticed there is no horizontal shift, but there is a vertical shift of 9 units down from the origin.

• The coefficient of \( (x-3)^2 \) is positive, which confirms the parabola opens upward.
• The vertex form helps easily locate the vertex of a parabola.
• The parabolas' symmetry line is the vertical line \( x=3 \).

Graphing becomes straightforward once these characteristics are recognized, allowing us to sketch more accurately.

Eliminating parameters involves transforming a parametric equation into a Cartesian equation. In this process, our goal is to derive a direct relationship between \( x \) and \( y \) without involving the parameter \( t \).
In the given problem, the parametric equations are \( x=4t+3 \) and \( y=16t^2-9 \). We start by isolating \( t \) from one of the equations. From \( x=4t+3 \), solve for \( t \) to get \( t = \frac{x-3}{4} \).
By substituting this expression in place of \( t \) in the equation for \( y \), we eliminate the parameter. This substitution gives us the equation \( y = 16\left( \frac{x-3}{4} \right)^2 - 9 \).
Simplifying further, we arrive at \( y = (x-3)^2 - 9 \), successfully eliminating \( t \) and finding the Cartesian equation representing the curve. The resulting equation helps in sketching the curve easily, as it resembles a familiar parabolic form.

Sketching a curve from an equation involves mapping the set of points that satisfy the relationship between \( x \) and \( y \). In our case, after eliminating the parameter, we are left with the equation \( y = (x-3)^2 - 9 \).
This equation represents a parabola with its vertex at \((3, -9)\). To sketch this curve, start by plotting the vertex. It's critical to remember that the parabola is symmetrical about the line \( x = 3 \).

• Plot several points on either side of the vertex to determine the curvature.
• At \( x = 4 \), substitute into the equation to find \( y = -8 \), providing another point for accuracy.
• Increment \( x \) to the left and right and calculate corresponding \( y \) values to get more points.

The overall shape should be an upward-opening curved "U" shape. Being aware of your parabola's direction, symmetry, and vertex location will help ensure a correct and precise sketching.