Problem 51 - Algebra 2

Question

Graph each equation. $$ (y-2)^{2}=4(x+3) $$

Step-by-Step Solution

Answer

The parabola with the equation \((y-2)^2 = 4(x+3)\) has a vertex at \((-3, 2)\), a focus at \((-2, 2)\), and opens to the right.

Step 1: Identify the vertex

The vertex of the parabola is at the point \((h, k)\). Here, \(h = -3\) and \(k = 2\). Therefore, the vertex of the given parabola is at \((-3, 2)\).

Step 2: Identify the value of \(p\)

The given equation is in the form \((y-2)^2 = 4(x+3)\), which can be rewritten as \((y-2)^2 = 4*1*(x+3)\). Comparing this with the general form of the parabola equation, \((y-k)^2 = 4p(x-h)\), we find that \(p = 1\).

Step 3: Determine the direction of the parabola

The sign of \(p\) determines the direction in which the parabola opens. If \(p > 0\), the parabola opens to the right. If \(p < 0\), it opens to the left. Here, \(p = 1\), which is greater than 0. Therefore, the parabola opens to the right.

Step 4: Identify the focus

The focus of a parabola is the point \((h+p, k)\). Substituting \(h = -3\), \(k = 2\), and \(p = 1\), we get the focus as \((-3+1, 2) = (-2, 2)\).

Step 5: Sketch the graph

Plot the vertex point \((-3, 2)\) and focus \((-2, 2)\) on the coordinate plane, and draw an arrow to the right from the vertex to indicate the opening of the parabola. Then, sketch the graph of the parabola using the vertex and focus to guide you.

Key Concepts

Vertex

In the context of parabolas, the vertex is a crucial point that essentially acts as the turning point or the 'tip' of the parabola. When you are dealing with a parabolic equation, the vertex gives you valuable information about the graph's location and orientation.
The equation provided, \((y-2)^2 = 4(x+3)\), is in a form called the vertex form of a parabola, \((y-k)^2 = 4p(x-h)\).

• Here, \(h = -3\)
• \(k = 2\)

Thus, the vertex is located at \((-3, 2)\).
This point is significant because the entire parabola is symmetrical around the line that passes through the vertex.

Focus

The focus of a parabola is another critical point, directly influencing its shape and size. Parabolas are defined by their geometric property where every point is at the same distance from the focus and a specific line called the directrix.
Understanding the relationship between the vertex and the focus can help you graph the parabola accurately.
The focus \((h + p, k)\) is \((-3 + 1, 2)\), which results in the point \((-2, 2)\).
Here's why the focus matters:

• The parabola "bends" around this point.
• It helps determine how "wide" or "narrow" the parabola is.

By knowing the location of the focus, you can tell how the parabola will graphically appear on a coordinate plane.

Direction of Parabola

The direction in which a parabola opens is guided by the sign and position of the parameter \(p\) in its equation. With our given equation, \((y-2)^2 = 4(x+3)\), it's clear to derive \(p = 1\) after comparing it to the general form \((y-k)^2 = 4p(x-h)\).
In this case:

• If \(p > 0\), the parabola opens to the right.
• If \(p < 0\), it opens to the left.

Since \(p = 1\) (a positive number), our parabola opens to the right.
This direction is essential for accurately sketching or visualizing how the parabola should be drawn on a graph.

Graphing Equations

Graphing equations accurately involves a series of steps where understanding the vertex, focus, and direction all play a part. Given the equation \((y-2)^2 = 4(x+3)\), one must start with identifying the vertex and focus for plotting.
Step-by-step:

• Plot the vertex at \((-3, 2)\).
• Identify and plot the focus at \((-2, 2)\).

The distance between the vertex and the focus determines the stretch of the parabola.
Remember:

• The parabola should open towards the direction derived, here to the right, due to \(p\).

Using the vertex and focus as guides, draw the parabolic curve from the vertex, ensuring it envelops the focus in a smooth, symmetrical shape.