Problem 44 - Algebra 2

Question

CHALENGE The parabola with equation \(y=(x-4)^{2}+3\) has its vertex at \((4,3)\) and passes through \((5,4) .\) Find an equation of a different parabola with its vertex at \((4,3)\) and that passes through \((5,4) .\)

Step-by-Step Solution

Answer

\( y = -(x-4)^2 + 3 \) is a different parabola with the same vertex and passing through (5,4).

Step 1: Identifying Original Parabola

The original parabola has the equation \( y = (x-4)^2 + 3 \), with vertex at \((4,3)\) and passes through the point \((5,4)\). This form is known as the vertex form of a parabola.

Step 2: General Vertex Form for New Parabola

To find a new parabola with vertex \((4,3)\) that passes through \((5,4)\), use the general vertex form: \( y = a(x-h)^2 + k \) where \( (h,k) = (4,3) \). This simplifies to \( y = a(x-4)^2 + 3 \).

Step 3: Using Given Point to Determine 'a'

Substitute the point \((5,4)\) into the equation \( y = a(x-4)^2 + 3 \):\[ 4 = a(5-4)^2 + 3 \]\[ 4 = a imes 1 + 3 \]\[ a = 4 - 3 = 1 \]

Step 4: Write New Parabola Equation

Using the calculated value of \( a = 1 \), the equation of the new parabola is:\[ y = (x-4)^2 + 3 \]

Step 5: Check for Different Parabola Condition

To ensure it is a different parabola, we can choose another value for \( a \) different from 1, for example, \( a = -1 \):\[ y = -(x-4)^2 + 3 \]. This new equation will have the same vertex \((4,3)\) and still pass through point \((5,4)\) if recalculations ensure consistency.

Key Concepts

Vertex Form

The vertex form of a parabolic equation provides a very intuitive view of the parabola. It is written as \( y = a(x-h)^2 + k \). Here, \( a \), \( h \), and \( k \) determine the properties and position of the parabola. Unlike standard forms, the vertex form directly highlights the vertex \((h, k)\). This form is handy for graphing because you can quickly spot the vertex just by looking at the equation. It is also helpful for transformations, as the parameter \( a \) controls the direction and width of the parabola. If \( a \) is positive, the parabola opens upwards, and if negative, it opens downwards. The absolute value of \( a \) affects how "narrow" or "wide" the parabola appears.

Coordinates

Understanding coordinates is crucial in geometry as it describes the position of points. In a 2D plane, a point is identified by two numbers, \( x \) and \( y \), written as \((x, y)\). The first number, \( x \), indicates the horizontal position relative to the origin, while \( y \) denotes the vertical position. Coordinates are not just point identifiers; they are powerful tools for identifying relationships between different elements on a plane. They are essential for graphing lines, parabolas, or any other geometric shapes. For instance, in the exercise, we use points \((4,3)\) and \((5,4)\) to find and confirm the equation of a parabola.

Vertex

The vertex of a parabola is a significant point as it is the point where the curve changes direction. In the vertex form \( y = a(x-h)^2 + k \), the vertex is easily identified at \((h, k)\). For a standard parabola, when \( a > 0 \), the vertex is the lowest point, and when \( a < 0 \), it is the highest. These characteristics make the vertex crucial for sketching parabolas effectively. The vertex also represents the axis of symmetry of the parabola, meaning the parabola is a mirror image on either side of the vertical line through the vertex.

Point

A point in geometry is a precise location in space. It is usually represented using coordinates, such as \((x, y)\) in a 2D plane. In parabolas, knowing specific points is necessary for identifying and sketching the curve. A single point, like the vertex \((4,3)\), can define the position and orientation of the parabola entirely. Other points, like \((5,4)\), allow us to adjust the parabola by affecting its shape when substituting them into the vertex form equation. By recalculating the equation with various points, different parabolic curves can be explored while keeping the vertex constant.