Problem 64 - Algebra and Trigonometry

Question

In each exercise, graph the equation in a rectangular coordinate system. $$y^{2}=4(x+1)$$

Step-by-Step Solution

Answer

The graph of the given equation should be a parabola that opens to the right with a vertex at (-1,0), focus at (0,0) and directrix at \(x = -2\).

Step 1: Arrangement to Standard Form

Rewrite the given equation \(y^{2} = 4(x+1)\) to the standard form of parabolic equations that is \(y^{2} = 4p(x - h)\). Since we do not have 'h' in our equation it gives us \(y^{2} = 4p(x - 0)\) which implies 'h' is zero (h=0). Also from the equation we can see that \(4p=4\) which implies that \(p=1\).

Step 2: Determine the Vertex

The vertex form of the equation for a parabola that opens right or left is \((x - h)^2 = 4p(y - k)\). The value of 'h' and 'k' are the x, y coordinates of the vertex of the parabola respectively. Therefore, looking at the equation \(y^{2} = 4p(x - 0)\), the vertex will be (h, k) = (-1, 0).

Step 3: Determine the Focus and Directrix

For a right or left opening parabola, the focus is given by (h+p, k). Hence, \(h+p = -1+1 = 0\). Therefore, the focus is at (0,0). The equation of the directrix for a right or left opening parabola is given by \(x = h - p\). Plugging h = -1 and p = 1 into the equation gives us x = -2.

Step 4: Plotting the Parabola

The graph should be a parabola that opens to the right. Firstly plot the vertex at (-1, 0). The focus of the parabola will be at the point (0, 0) and the directrix is the line \(x = -2\). The graph of the parabola has a vertex at (-1, 0), opens to the right, because p is positive and 4p = 4. The width of the parabola at the vertex is 2p. Hence in this case it is 2.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system is like a large grid, often used in mathematics to plot graphs and equations. It consists of two lines that intersect at right angles. These lines are known as the x-axis and the y-axis.

• X-axis: This is the horizontal line that runs left to right.
• Y-axis: This is the vertical line that runs up and down.
• Point of Intersection: The point where these two lines meet is labeled as the origin, typically marked as (0,0).

When you graph anything using this system, each point on the graph corresponds to an ordered pair. An ordered pair is written as (x, y), where 'x' tells you how far to move on the x-axis and 'y' tells you how far to move on the y-axis.
This system helps to clearly visualize and interpret the solutions of equations, like the parabola in the exercise.

Vertex of a Parabola

A parabola is a U-shaped graph with a unique point known as the vertex. The vertex is the highest or lowest point on the parabola, depending on its direction.

• In our example, the equation is in the form of a sideways parabola, given by \(y^2 = 4p(x - h)\), where (h, k) is the vertex.
• By comparing the equations, we find that (h, k) becomes (-1, 0) for our parabola.
  
• The vertex here at (-1, 0) means that the parabola starts "turning" at this specific point.

Understanding the vertex is crucial because it serves as a reference point for the direction and position of the entire parabola on the grid.

Focus and Directrix of a Parabola

The focus and directrix are key elements tied to the shape and position of a parabola.

• Focus: This is a point inside the parabola where it is always aimed. In our example, the focus is found at (0, 0).
• Directrix: This is a line outside the parabola that is perpendicular to the parabola's axis of symmetry. For our parabola, it's the line \(x = -2\).
• All points on a parabola are equidistant from the focus and the directrix, meaning a balance between the two.

By identifying these parts, one can better understand how the parabola will open and the nature of its curvature.

Standard Form of Parabolic Equations

Parabolic equations can usually be reorganized into a standard form, which simplifies graphing and analyzing them. For a parabola that opens left or right, the standard form is \(y^2 = 4p(x - h)\).

• Here, 'h' and 'k' represent the coordinates of the vertex.
• 'p' is a crucial parameter because it tells us the distance from the vertex to the focus along the axis of symmetry.
• In the exercise equation \(y^2 = 4(x + 1)\), comparing with the standard form gave us 'p' as 1, showing the parabola opens to the right.

This form makes it easier to spot key features like the vertex, focus, and direction of the parabola just by looking at the equation.