Problem 35 - College Algebra

Question

Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. $$ (x-2)^{2}=8(y-1) $$

Step-by-Step Solution

Answer

vertex = (2,1), focus = (2,3), directrix = y=-1.

Step 1 - Identify the values of h, k and p

In the given equation \((x-2)^{2}=8(y-1)\), comparing with the standard form, we can deduce that \(h=2\), \(k=1\) and \(4p=8\). Thus, \(p=8/4=2\).

Step 2 - Find the vertex, focus and directrix

The vertex of the parabola is given by the point \((h,k)\), which is \((2,1)\). \n The focus of the parabola is \((h, k+p)= (2,1+2)= (2,3)\) since the parabola opens upwards. \n Since the parabola is vertical and opens upwards, the directrix is a horizontal line given by the equation \(y=k-p\), which is \(y=1-2=y=-1\).

Step 3 - Graph the parabola

Draw the vertex point, the focus point and the directrix line on the graph. The parabola opens upwards, and the point where it turns is the vertex point \((2,1)\). The line y = -1 is below the vertex since the parabola opens upwards. The parabola curves upwards towards the focus point (2,3).

Key Concepts

Vertex

In parabolas, the vertex is the point where the curve changes direction. Imagine it as the peak or the "turn-around" point. When we look at the equation \((x-2)^2 = 8(y-1)\), we can match it to the standard form \((x-h)^2 = 4p(y-k)\). This helps us find the vertex. Here, \(h = 2\) and \(k = 1\). Hence, our vertex is at the point \((2, 1)\). It acts like a starting point for graphing, telling us where the parabola sits on the graph.

• Vertex Formula: \((h, k)\)
• For \((x-2)^2 = 8(y-1)\), the vertex is \((2, 1)\)

The vertex is crucial because it provides a central point from which the parabola expands.

Focus

The focus of a parabola is a special point that helps define its shape. Imagine it as a beacon that the parabola "listens to," guiding its path. From the equation \((x-2)^2 = 8(y-1)\), we've determined that \(p = 2\). And since the parabola opens upwards, the focus will be above the vertex. We calculate it with the formula \((h, k+p)\). So, for our example, the focus lies at \((2, 3)\).

• Focus Formula: \((h, k+p)\)
• For our equation, focus is \((2, 3)\)

Understanding the focus is key to visualizing how steep or wide your parabola will appear.

Directrix

A directrix is a line that's equally as important as the focus when it comes to defining a parabola’s shape. Instead of a point, it's a straight horizontal line when a parabola opens upward or downward. In our scenario, since the parabola opens upwards, the directrix is found below the vertex. We use the formula \(y = k - p\). Plugging in our values gives us the equation \(y = 1 - 2 = -1\). So, the directrix of the parabola is the line \(y = -1\).

• Directrix Formula: \(y = k - p\)
• For our parabola, directrix is \(y = -1\)

The directrix works in tandem with the focus to provide guiding limits for the parabola’s curve formation.

Graphing Parabolas

Graphing a parabola means plotting out its vertex, focus, and directrix onto a coordinate plane. These focal points and lines act like a blueprint for sketching the curve. Start with the vertex at \((2, 1)\), and mark it clearly on the graph. Next, place the focus at \((2, 3)\), roughly indicating where the curve will aim towards. Graph the directrix as a line at \(y = -1\). The parabola will curve upwards in this scenario, like a smile, starting from the vertex and heading toward the focus.

• Start with vertex: \((2, 1)\)
• Place focus: \((2, 3)\)
• Draw directrix line: \(y = -1\)

If you visualize how these elements work together, sketching the actual parabola becomes clear and simple.