Problem 13 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The vertex is at (3, 2).

Step 1: Understand the Equation

The given equation is \( x = (y - 2)^2 + 3 \). This is in the form of a parabola opening sideways because it's solving for \( x \) in terms of \( y \). The standard form for a sideways opening parabola is \( x = (y-k)^2 + h \), where \( (h, k) \) is the vertex.

Step 2: Identify the Vertex

In the equation \( x = (y - 2)^2 + 3 \), it is clear that \( h = 3 \) and \( k = 2 \). Therefore, the vertex of the parabola is at the point \( (3, 2) \).

Step 3: Graph the Parabola

To graph the parabola, plot the vertex at \( (3, 2) \). Since the expression \( (y-2)^2 \) is squared, the parabola will open to the right. Plot additional points to the left and right of the vertex to accurately draw the parabola.

Key Concepts

graphing parabolas

Graphing a parabola can seem like a daunting task, but breaking it down makes it more approachable. Parabolas come in different shapes and orientations, mainly based on how they open. They can open upwards, downwards, or sideways. The most critical element of a parabola's graph is its vertex, which is the highest or lowest point, depending on the orientation.
To graph a parabola:

• Identify the vertex, which is key in plotting.
• Know the direction in which the parabola opens (up, down for vertical; left, right for horizontal).
• Plot the vertex on the graph first.
• Add other points symmetrically around the vertex to define its shape.

Once you have your set of points, you can connect them to outline the parabola's shape. Remember, the symmetry of the parabola makes graphing easier as one side mirrors the other.

sideways parabola

A sideways parabola differs from the usual 'U' shaped one that opens up or down. Here, it opens left or right. These types of parabolas have the equation solved for \(x\) instead of \(y\). Essentially, in its standard form, it looks like \(x = (y-k)^2 + h\).
Such form indicates that the parabola will open horizontally:

• If the term in front of the squared part is positive, it opens to the right.
• If negative, the parabola opens to the left.

In the equation \(x = (y - 2)^2 + 3\), we have a sideways parabola opening to the right! Understanding this peculiar nature helps in plotting and analyzing these curves correctly. As always, identify the vertex first to start your graph.

finding vertex of parabola

Finding the vertex of a parabola is one of the first steps when graphing. The vertex acts like a pivot point around which the rest of the graph is structured. For normal parabolas, the form \(y = ax^2 + bx + c\) will have the vertex at \(x\) calculated using \(-b/2a\).
However, for sideways parabolas like \(x = (y-k)^2 + h\), the vertex is easy to spot. It's at the point \((h, k)\) based on comparison with the standard form.
In simpler terms:

• \

finding vertex of parabola

Find \(h\) from the constant outside the square, which also dictates the shifted side.

Find \(k\) inside the parentheses with the opposite sign (e.g., \(y - 2\) means \(k = 2\)).

Combine \(h\) and \(k\) to determine the vertex position \((h, k)\).

For our example, \(x = (y-2)^2 + 3\), the vertex is plainly \((3, 2)\). This point is fundamental for drawing the entire parabola and understanding its orientation.