Problem 45 - Algebra 2

Question

Write an equation for each conic section. Then sketch the graph. parabola with vertex \((-3,2)\) and focus \((-3,7)\)

Step-by-Step Solution

Answer

The equation of the parabola with vertex \((-3,2)\) and focus \((-3,7)\) is \((x + 3)^2 = 20(y - 2)\). The graph of this parabola opens upwards, with vertex at \((-3, 2)\) and focus at \((-3, 7)\), and symmetric about the line \(x = -3\).

Step 1: Identify the given

From the problem, the vertex \(V\) is given as \(-3, 2\), so \(h = -3\) and \(k = 2\), and the focus \(F\) is given as \(-3, 7\), so the coordinate of the focus is \(-3, 7\). This means that the parabola opens upwards.

Step 2: Calculate the focal distance

To find the value of \(p\), we can use the distance formula which is the difference in the \(y\)-values of the vertex and the focus. Hence, \(p = 7 - 2 = 5\).

Step 3: Write the equation of the parabola

We can now substitute the values of \(h\), \(k\), and \(p\) into the equation of the parabola. Therefore the equation is \((x - (-3))^2 = 4*5(y - 2)\). Hence the equation of the parabola is \((x + 3)^2 = 20(y - 2)\).

Step 4: Sketch the graph

To sketch the graph, first draw the parabola's axis of symmetry, which is the vertical line \(x = -3\). Mark the vertex \(V(-3, 2)\) and the focus \(F(-3, 7)\). Draw a line called the directrix five units below the vertex (since the focal distance \(p = 5\)), i.e., the line \(y = -3\). Now, sketch the parabola opening upwards towards the focus, with vertex \(V\) and symmetric with respect to the line \(x = -3\).

Key Concepts

Parabola

A parabola is a type of conic section, recognizable by its characteristic U-shape. It consists of all the points that are equidistant from a specific point known as the focus and a line known as the directrix.

Parabolas can open upwards, downwards, left, or right. The direction they open depends on the orientation of their vertex and focus. Each parabola is defined by a unique equation based on its position and orientation. When you study parabolas, you'll typically encounter equations in the form \((x-h)^2 = 4p(y-k)\) for vertical parabolas, or \((y-k)^2 = 4p(x-h)\) for horizontal parabolas.

• 'h' and 'k' are the coordinates of the vertex.
• 'p' is the distance from the vertex to the focus.

Vertex

The vertex of a parabola is its highest or lowest point, depending on its orientation. It is the point where the parabola changes direction. For the given problem, the vertex is located at \((-3, 2)\).

The vertex is crucial in forming the parabola's equation. In standard form, the vertex is the point \((h, k)\) in the equations, \((x-h)^2 = 4p(y-k)\) for upward or downward opening, and \((y-k)^2 = 4p(x-h)\) for leftward or rightward opening.

• The vertex acts as a symmetrically central point for the parabola.
• Knowing the vertex helps in graphing the curve accurately.

Focus

The focus of a parabola is a point from which distances to any point on the parabola are measured, parallely to the directrix. For this exercise, the focus is given as \((-3, 7)\).

The focus is encircled by the parabolic arms, shaping the curve's inward bow, directing towards it. The focus along with the directrix determines the shape and width of the parabola.

• It determines the direction in which the parabola opens.
• The focal length, denoted as 'p', is the distance from the vertex to the focus.

The sharper a parabola's curve, the closer its focus and vertex are.

Graphing Equations

Graphing a parabola's equation involves several steps to visualize its shape and orientation correctly. Begin by identifying the vertex and focus from the given information; for the parabola in this exercise, focus on the vertex at \((-3, 2)\).

Next, calculate the focal distance, denoted as 'p', which is the distance between the vertex and the focus. Here, the focal distance \(p = 5\), indicating that the parabola stretches upwards five units from the vertex.

• Start by sketching the axis of symmetry, influenced by the vertical or horizontal nature of the parabola. For this exercise, the axis of symmetry is the line \(x = -3\).
• Mark the vertex and draw the focus.
• Also draw the directrix, which for upward or downward parabolas like this one, lies symmetrically opposite the focus.

Finally, sketch the parabola forming a smooth curve from the vertex, encircling the focus, with its arms approaching but never touching the directrix.