Problem 104 - A Graphical Approach to Precalculus with Limits

Question

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$(y+2)^{2}=x+1$$

Step-by-Step Solution

Answer

Vertex: (-1, -2), Axis: y = -2, Domain: [-1, ∞), Range: (-∞, ∞).

Step 1: Write the equation in standard form

The given equation \[(y + 2)^2 = x + 1\] can be rearranged as \[x = (y + 2)^2 - 1\].Here, this is a sideways parabola with the standard form \(x = a(y - k)^2 + h\), where \(h\) and \(k\) define the vertex.

Step 2: Identify the vertex and axis

In the equation \(x = (y+2)^2 - 1\), compare it with \(x = a(y - k)^2 + h\), we identify the vertex \((h, k) = (-1, -2)\). The axis is a horizontal line \(y = -2\) since it is a sideways parabola.

Step 3: Determine the domain

Because the parabola opens to the right (positive direction of the x-axis), the domain is all real numbers greater than or equal to the x-coordinate of the vertex. Thus, the domain is \([ -1, \infty)\).

Step 4: Determine the range

Since the parabola covers all values of \(y\) as it extends infinitely in both directions along the \(y\)-axis, the range is \((-\infty, \infty)\).

Step 5: Graph the parabola

To graph this sideways parabola, plot the vertex at \((-1, -2)\). Because the parabola opens to the right, sketch a curve opening from \((-1, -2)\) extending towards the positive x-axis. Use a few points by substituting values of \(y\) around -2 to aid in sketching.

Step 6: Verify with a graphing calculator

Enter the function into the graphing calculator, ensuring the equation is rearranged to monitor a shape consistent with the manually sketched graph. It should verify that the vertex and axis are correct, and the curve matches.

Key Concepts

Vertex of a Parabola

The vertex is a crucial feature of a parabola. It represents the turning point of the graph, where the direction changes. For our equation, \[ x = (y + 2)^2 - 1 \]we identify the vertex by comparing this with the standard form of a sideways parabola: \[ x = a(y - k)^2 + h \]Here, \( h = -1 \) and \( k = -2 \), so the vertex is \((-1, -2)\).

The vertex gives us a fixed point on the graph that helps in sketching the parabola. In the graph of a sideways parabola like this one, the vertex is at the minimum or maximum x-value, depending on the parabola's direction. The parabola 'opens' from this point, extending in a horizontal direction.

Axis of Symmetry

For parabolas, the axis of symmetry is the line that runs through the vertex, dividing the parabola into two mirror-image halves. It is very helpful in graphing because it assures us that both sides of the parabola are symmetrical. For our sideways parabola, the axis of symmetry appears as a horizontal line.

If the parabola is written in the form \[ x = a(y - k)^2 + h \]like our equation, the axis of symmetry is given simply by the value of \( y \) from the vertex, which is \( y = -2 \).

• Gives a straight reference line to check the shape of the parabola.
• Ensures symmetry about this line for an accurate graph.

Domain and Range

Understanding domain and range is key to fully grasping how a given parabola behaves. The domain of a parabola refers to all possible x-values the function can take on, while the range refers to all possible y-values.In our exercise, since the parabola opens to the right, the domain includes all real numbers starting from the x-coordinate of the vertex: \[ [-1, \infty) \].This means it stretches infinitely to the right on the x-axis.

The range, on the other hand, covers all real numbers from negative infinity to positive infinity because the parabola extends vertically without bound: \((-\infty, \infty)\).

• Domain: All x-values the parabola can cover, giving us its horizontal spread.
• Range: All y-values it includes, which in this case is unrestricted both ways vertically.