Problem 42 - Algebra 2

Question

OPEN ENDED Write an equation for a parabola that opens to the left. Use the parent graph to sketch the graph of your equation.

Step-by-Step Solution

Answer

The equation is \( x = -(y-2)^2 + 3 \).

Step 1: Understand Parabola Orientation

A parabola has different forms depending on its orientation. For a parabola that opens to the left, the graph is a horizontal parabola. The equation of a horizontal parabola is generally given by \( x = a(y - k)^2 + h \), where \((h, k)\) is the vertex. The sign of \(a\) determines the direction; if \(a < 0\), the parabola opens to the left.

Step 2: Choose Vertex and Parameter

Select a vertex for the parabola, for instance, \((h, k) = (3, 2)\). Choose a value for \(a\) to determine how wide or narrow the parabola is. Let \(a = -1\), making the equation \( x = -(y - 2)^2 + 3 \). Here, the negative sign ensures it opens to the left.

Step 3: Verify the Equation

Check that the chosen equation, \( x = -(y - 2)^2 + 3 \), meets the criteria. Here, the vertex of the parabola is \((3, 2)\), and the parabola opens to the left because \(a = -1\). If these conditions are met, the equation fulfills the requirement for a left-opening parabola.

Step 4: Sketch the Graph

To sketch the graph, plot the vertex \((3, 2)\) on a coordinate grid. Since the parabola opens to the left and \(a = -1\), draw a curved line that opens leftward and passes symmetrically through the vertex. The width and steepness are adjusted based on the value of \(a\), with \(a = -1\) indicating a standard width.

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is a specific way to express the equation of a parabola, which emphasizes the vertex's location. Unlike the standard quadratic equation, vertex form makes it easy to identify the turning point, or vertex, of the parabola. For a horizontal parabola, the vertex form is expressed as:

• \( x = a(y - k)^2 + h \)

Here, \((h, k)\) are the coordinates of the vertex. This form is very convenient because it clearly tells us where the parabola will change direction. The parameter \(a\) in the equation affects the width and direction of the parabola. A positive \(a\) makes it pass to the right, whereas a negative \(a\) directs it to open to the left. The vertex form makes graphing and understanding the parabolic curves much more manageable.

Graphing Parabolas

Graphing parabolas involves plotting the parabola on a coordinate plane to visualize its shape and orientation. When using the vertex form of a horizontal parabola, it begins with identifying the vertex \((h, k)\) on the graph. This point is crucial because it is the center of symmetry for the parabola.
Next, the value of \(a\) should be considered to understand the steepness and direction.

• If \(a > 0\), the parabola opens to the right.
• If \(a < 0\), the parabola opens to the left.

The smaller the absolute value of \(a\), the wider the parabola. Conversely, a larger absolute value of \(a\) results in a narrower curve. When graphing, ensure the parabola is correctly symmetrical around the axis passing through the vertex and follows the direction defined by \(a\).

Parabola Orientation

The orientation of a parabola is determined mainly by the sign of the parameter \(a\) in the vertex form equation. For horizontal parabolas, which are

• \( x = a(y - k)^2 + h \)

The orientation decides whether the parabola opens left or right.

• A left or right orientation in horizontal parabolas depends largely on whether \(a\) is negative or positive, respectively.
• This is opposed to vertical parabolas where the orientation is determined by a more traditional "+ up, \- down" mechanic.

Because of this particular property, horizontal parabolas require extra attention to avoid mistakes in predicting direction and graph alignment.

Left-Opening Parabolas

Left-opening parabolas are a specific type of horizontal parabola where the curves bend towards the negative x-direction. This particular orientation occurs when the value of \(a\) in the parabola's vertex form equation is negative. For the general formula:

• \( x = a(y - k)^2 + h \)

If \(a < 0\), then the parabola opens to the left.
Understanding the properties of left-opening parabolas is crucial in graphing exercises:

• The vertex \((h, k)\) remains a fixed point of symmetry and helps plot the curve accurately.
• Negative values widen or narrow the parabola, creating different shapes on the graph.

Thus, accurately recognizing and calculating the effect of \(a\) ensures correct depiction and understanding of these unique parablolic graphics, especially within practical applications.