Problem 39 - Thomas Calculus

Question

You may wish to review Section 1.2 before solving Exercises \(39-56\) The parabola \(y^{2}=8 x\) is shifted down 2 units and right 1 unit to generate the parabola \((y+2)^{2}=8(x-1)\) . $$ \begin{array}{l}{\text { a. Find the new parabola's vertex, focus, and directrix. }} \\ {\text { b. Plot the new vertex, focus, and directrix, and sketch in the }} \\ {\text { parabola. }}\end{array} $$

Step-by-Step Solution

Answer

Vertex: (1,-2), Focus: (3,-2), Directrix: x=-1.

Step 1: Identify the original parabola and transformation

The original equation of the parabola is given by \( y^2 = 8x \). The problem states that the parabola is shifted down 2 units and right 1 unit to produce the new equation \( (y+2)^2 = 8(x-1) \). This transformation is a translation of the original graph.

Step 2: Find the new vertex of the parabola

The vertex of the parabola \( y^2 = 8x \) is at the origin \((0,0)\). After the transformation (right 1, down 2), the new vertex is at \((1,-2)\).

Step 3: Determine the new focus of the parabola

For the original parabola \( y^2 = 8x \), the focus is \(\left(\frac{a}{4}, 0\right) = (2, 0)\), where \( a = 8 \). After applying the same translation, right 1 and down 2, the new focus is at \((3, -2)\).

Step 4: Find the new directrix of the parabola

The original directrix for \( y^2 = 8x \) is \( x = -2 \). Applying the same transformation (right 1 unit), the new directrix is \( x = -1 \) (the downward transformation doesn't affect the x-coordinate).

Step 5: Plot the vertex, focus, and directrix

To plot the new parabola, you need to locate the points: the vertex at \((1, -2)\), the focus at \((3, -2)\), and draw the line for the directrix \( x = -1 \). The parabola will open to the right, with the vertex as its starting point.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a key point that offers valuable information about the parabola's position. Think of it as the point where the parabola changes direction. For a parabola given in the form \[(y - k)^2 = 4p(x - h)\]the vertex is located at the point \((h, k)\).

• In our exercise, the original parabola is \(y^2 = 8x\). Its vertex starts at the origin \((0, 0)\).
• Upon transformation—moving right by 1 and down by 2—the new vertex shifts to \((1, -2)\).

This shift is due to the translation of the graph based on the new coordinates within the parentheses. The vertex gives us an anchor point from which the shape of the parabola can be plotted. In this case, it helps establish the direction and path of the opening.

Focus of a Parabola

The focus of a parabola is another fundamental element. It is a fixed point that, along with the directrix, helps define the parabola. Every point on the parabola is equidistant from the focus and the directrix. In an equation such as \((y - k)^2 = 4p(x - h)\), the focus is given by the coordinates \((h + p, k)\). For the original parabola \(y^2 = 8x\), the focus is located at \((2, 0)\). Here, \(p\) corresponds to \(a/4\), where \(a\) is the coefficient from the equation.

• Once the entire graph is translated right by 1 and down by 2, the focus moves to \((3, -2)\).

This point remains inside the curve of the parabola. Knowing the focus is essential, as it helps in sketching the parabola accurately and understanding its geometric properties.

Directrix of a Parabola

The directrix of a parabola is a line that plays a crucial role along with the focus in defining the parabola's shape. For the same parabolic equation, \((y - k)^2 = 4p(x - h)\), the directrix can be found with the line equation \(x = h - p\). In the case of the original parabola \(y^2 = 8x\), its directrix is \(x = -2\). This line runs parallel to the axis of symmetry of the parabola, opposite to the direction of the focus.

• After the translational shift in this exercise, the directrix moves to \(x = -1\).

The directrix helps in determining the set of all points that form the parabola, through their equidistance from both the focus and the directrix. By plotting this line, one can visualize how the parabola opens and the space it occupies.