Problem 7 - Algebra A Combined Function

Question

The graph of each equation is a parabola. Find the vertex of the parabola and then graph it. See Examples 1 through 4. $$ x=(y-2)^{2}+3 $$

Step-by-Step Solution

Answer

Vertex: \((3, 2)\)

Step 1: Identify the Parabola Type

The given equation is in the form of \(x=(y-k)^2+h\), which is a parabola that opens sideways (left or right). This equation format represents a parabola that opens to the right when \((y-k)^2\) is positive and to the left when it is negative. Here, \((y-2)^2\) is positive, so the parabola opens to the right.

Step 2: Find the Vertex

For a parabola in the form \(x=(y-k)^2+h\), the vertex is located at the point \( (h, k) \). In the equation \(x=(y-2)^2+3\), \(h=3\) and \(k=2\). So, the vertex of the parabola is at \((3, 2)\).

Step 3: Graph the Parabola

Plot the vertex at point \((3, 2)\) on the graph. Since the parabola opens to the right, for each unit you move up or down from the vertex (in the y-direction), you will move further right in the x-direction by the square of that unit difference. Sketch the parabola starting from the vertex \((3,2)\), bulging towards the right.

Key Concepts

Parabola Opening Direction

Understanding the direction in which a parabola opens is essential for graphing it correctly. Parabolas can either open upwards, downwards, to the left, or to the right, depending on the equation format.
For a standard quadratic equation in the form \(y = ax^2 + bx + c\), the parabola will open upwards if \(a > 0\) and downwards if \(a < 0\). However, for parabolas that open sideways, the equation takes the form \(x = (y-k)^2 + h\).
In this format, the parabola opens to the right if the squared term \((y-k)^2\) is positive and to the left if it is negative. In the given exercise, the equation is \(x = (y-2)^2 + 3\). Since the term \((y-2)^2\) is positive, the parabola opens to the right.

• If \((y-k)^2\) is positive, the opening is to the right.
• If \((y-k)^2\) is negative, the opening is to the left.

Recognizing this pattern helps in sketching the correct graph of the parabola.

Vertex Form of a Parabola

The vertex form of a parabola provides a valuable way of expressing the equation, highlighting the vertex's position. This is particularly useful when graphing because it gives direct insight into the parabola's peak or lowest point.
For parabolas opening up or down, the vertex form is given by \(y = a(x-h)^2 + k\). For parabolas opening sideways, it is expressed as \(x = (y-k)^2 + h\). Here, the vertex is located at point \((h, k)\).
In our exercise with the equation \(x = (y-2)^2 + 3\), the vertex is easily identifiable as \((h, k) = (3, 2)\).

• The vertex \((h, k)\) gives the turning point of the parabola.
• Helps in quickly sketching the graph with the correct orientation and scale.

Understanding the vertex form simplifies the process of finding the parabola's shape and orientation on a graph.

Graphing Parabolas

Graphing parabolas involves careful plotting based on the equation's structure. Once you understand the parabola's opening direction and locate the vertex, sketching the graph becomes straightforward.
Start by plotting the vertex on the coordinate plane. In our case, this is the point \((3, 2)\). Next, decide the direction the parabola opens based on the equation format. Here, it opens to the right.
When graphing:

• Focus on symmetry around the vertex; for every point on one side, there's a mirror point on the other.
• Remember that moving away from the vertex, the parabola spreads out. This involves progressing x by the square of the y-units (both up and down from the vertex).
• Sketch smoothly, maintaining a consistent curve that doesn't abruptly change direction.

By laying down these foundational steps, you ensure the parabola is accurately depicted as intended by the given equation.