Problem 103 - 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. $$x=(y-3)^{2}$$

Step-by-Step Solution

Answer

The vertex is (0, 3), axis y = 3, domain [0, ∞), and range (−∞, ∞).

Step 1: Identify the Form of the Equation

The given equation is in the form of \( x = (y-k)^2 + h \), where \( k = 3 \) and \( h = 0 \). This form indicates a sideways parabola.

Step 2: Determine the Vertex

In the equation \( x = (y-3)^2 \), the vertex is \((h, k) = (0, 3)\). This is the point where the parabola changes direction.

Step 3: Find the Axis of Symmetry

For the equation \( x = (y-3)^2 \), the axis of symmetry is the line \( y = k \). Thus, the axis of symmetry is \( y = 3 \).

Step 4: Determine the Domain

Since the equation is in terms of \( x \) and involves a squared term, the possible values for \( x \) start from 0 and can go to infinity. Therefore, the domain is \( [0, \, +\infty) \).

Step 5: Determine the Range

The variable \( y \) is unrestricted as we don't have any limiting conditions. Thus, the range of \( y \) is all real numbers, \((-\infty, \, +\infty) \).

Step 6: Draw a Rough Sketch of the Parabola

To sketch the parabola, plot the vertex at \( (0, 3) \). The parabola opens to the right since it is in the form \( x = (y-3)^2 \). Sketch the curve opening to the right aligned with the axis of symmetry \( y = 3 \).

Step 7: Verification with a Graphing Calculator

After sketching the graph manually, input the equation \( x = (y-3)^2 \) into a graphing calculator to verify the sketch. Confirm that the vertex is at \( (0, 3) \) and the parabola opens to the right.

Key Concepts

Vertex

In the context of a parabola, the vertex refers to a crucial point where the parabola changes its direction. Imagine it as the peak or the lowest point of the shape, depending on its orientation.
In the given exercise, we have a sideways parabola described by the equation \(x = (y-3)^2\). Notice that the formula used is slightly different from what we usually see. Here, it's expressed in the form \(x = (y-k)^2 + h\), where the vertex is represented by the coordinates \((h, k)\).
The vertex for this particular equation is \((0, 3)\), because \(h = 0\) and \(k = 3\). This tells us that the point at \(0\) on the x-axis and \(3\) on the y-axis marks where the parabola changes direction. Understanding the vertex's location helps in sketching the parabola as well as understanding the symmetry and limits of the graph.

Axis of Symmetry

The axis of symmetry is an imaginary straight line that neatly divides a parabola into two identical halves. It acts almost like a mirror, showing that for every point on one side, there is an equal point on the other side, relative to it.
In our given equation, \(x = (y-3)^2\), the parabola is oriented sideways, meaning the symmetry is vertical rather than horizontal. The axis of symmetry corresponds to the "\(y\)" in the vertex form and is noted by the line \(y = k\).

• This means for our parabola, the axis of symmetry is \(y = 3\).
• Every point on the parabola remains equidistant on either side of this line, helping us predict the shape and behavior of the parabola easily.

This concept is crucial for correctly sketching and understanding the graph of a parabola, providing balance and alignment to the overall visual representation.

Domain and Range

The domain and range of a parabola tell us about the set of possible values that its coordinates can take.

• **Domain:** For our equation \(x = (y-3)^2\), this specifies which \(x\)-values the parabola can express. With this sideways parabola, the domain — possible \(x\) values — starts at the vertex and extends infinitely. Quite simply, the domain is \([0, \, +\infty)\).
• **Range:** Refers to the possible \(y\)-values the parabola can reach. Notice that there aren't restrictions on \(y\); hence, \(y\) values cover all real numbers, given as \((-\infty, \, +\infty)\).

The domain and range uniquely convey the potential span of the parabola, and understanding them is integral for accurately sketching the graph and for recognizing the limitations and extensions of the parabola in different applications.