Problem 103 - A Graphical Approach to Precalculus with Limits

Question

Graph by hand the equation of the circle or the parabola with a horizontal axis. $$x=y^{2}+1$$

Step-by-Step Solution

Answer

The parabola \( x = y^2 + 1 \) is graphed with vertex at \( (1, 0) \) and opens to the right.

Step 1: Identify the Equation Type

The equation provided is \( x = y^2 + 1 \). This is an equation of a parabola because it can be rewritten in the standard form \( x = a(y - k)^2 + h \), which identifies it as a parabola with a horizontal axis of symmetry.

Step 2: Rewrite in Vertex Form

The given equation \( x = y^2 + 1 \) is already in vertex form \( x = (y - 0)^2 + 1 \), where \( a = 1 \), \( h = 1 \), and \( k = 0 \). The vertex of the parabola is at \( (1, 0) \).

Step 3: Determine the Direction of the Parabola

Since \( a > 0 \) (\( a = 1 \)), the parabola opens to the right.

Step 4: Identify Key Points

Starting from the vertex \( (1, 0) \), select points on either side of the vertex to find additional points on the parabola. For example, when \( y = 1 \), \( x = (1)^2 + 1 = 2 \), giving the point \( (2, 1) \). Similarly, for \( y = -1 \), \( x = (-1)^2 + 1 = 2 \), giving the point \( (2, -1) \).

Step 5: Plot the Points and Draw the Parabola

Plot the vertex \( (1, 0) \) and the key points \( (2, 1) \) and \( (2, -1) \). Then draw a smooth curve through these points, extending the curve to the right to indicate the parabola opening that direction. Ensure the symmetry across the horizontal axis is visible.

Key Concepts

Parabola Vertex Form

The vertex form of a parabola is an essential way to understand and graph the curve efficiently. A parabola in vertex form appears as either \( y = a(x - h)^2 + k \) if it opens upwards or downwards, or \( x = a(y - k)^2 + h \) if it opens to the right or left. In this form:

• \( (h, k) \) represents the vertex of the parabola, the turning point of the curve.
• \( a \) dictates the width and the direction in which the parabola opens. A larger \( |a| \) value means a steeper parabola, while the sign of \( a \) tells whether it opens upwards or downwards (for \( y \) functions) and right or left (for \( x \) functions).

To graph a parabola given in an equation such as \( x = y^2 + 1 \), recognize that it resembles the form \( x = a(y - k)^2 + h \). Here,

• \( a = 1 \), meaning the parabola opens to the right and is neither too wide nor too narrow.
• \( h = 1 \), \( k = 0 \), indicating the vertex of the parabola is at the point (1, 0).

The vertex form allows for an intuitive plotting of the parabola on a coordinate plane starting at the vertex.

Axis of Symmetry

The axis of symmetry is a vital concept when studying parabolas. This is an imaginary line that runs through the vertex of the parabola, dividing it into two mirror-image halves. For a parabola in the form \( x = a(y - k)^2 + h \), the axis of symmetry is horizontal, unlike the usual vertical axis seen in \( y \)-function parabolas. In this case, the equation of the axis of symmetry is \( y = k \).

• In our example, the axis of symmetry is \( y = 0 \), as derived from the vertex form where \( k = 0 \).
• This line reflects every point on one half of the parabola precisely onto the other half.

Understanding the axis of symmetry aids in plotting the graph symmetrically and recognizing the inherent balance of the parabola's shape.

Direction of Opening

Determining the direction of a parabola's opening is a fundamental aspect of graphing it correctly. The coefficient \( a \) in the vertex form \( x = a(y - k)^2 + h \) or \( y = a(x - h)^2 + k \) will indicate this direction.

• If \( a > 0 \), a horizontal parabola opens to the right, whereas a vertical parabola opens upward.
• Conversely, if \( a < 0 \), a horizontal parabola opens to the left, while a vertical parabola opens downward.

In our specific parabola described by \( x = y^2 + 1 \), since \( a = 1 \), the parabola opens to the right.Recognizing this helps in visualizing the graph and predicting how it extends in the coordinate plane. Graphically, this means as you move further along the axis of symmetry from the vertex, the parabola spreads outward in the rightward direction, emphasizing the central placement of the vertex and symmetric structure.