Problem 78 - College Algebra with Modeling and Visualization

Question

Graph the parabola. $$ 1.4(y-1.5)^{2}=0.5(x+2.1) $$

Step-by-Step Solution

Answer

The parabola has vertex \((-2.1, 1.5)\) and opens to the right.

Step 1: Identify the Form of the Equation

Recognize that the given equation \( 1.4(y-1.5)^{2}=0.5(x+2.1) \) resembles the standard form \( (y-k)^2 = 4p(x-h) \) of a parabola that opens horizontally. Here, \( h = -2.1 \), \( k = 1.5 \), and \( 4p = \frac{0.5}{1.4} = \frac{5}{14} \).

Step 2: Determine the Vertex

The vertex of the parabola can be directly read from the standard form as \( (h, k) \). From the equation, the vertex is \( (-2.1, 1.5) \).

Step 3: Calculate the Value of p

Solve for \( p \) in the expression \( 4p = \frac{5}{14} \). This simplifies to \( p = \frac{5}{56} \), which indicates that the parabola opens to the right because \( p > 0 \).

Step 4: Identify the Direction of the Opening

Since \( p > 0 \), the parabola opens rightwards. If \( p < 0 \), it would open leftwards.

Step 5: Plot the Vertex and Additional Points

Plot the vertex \( (-2.1, 1.5) \) on a coordinate plane. Choose additional values for \( y \) around 1.5 and solve for corresponding \( x \) values using the equation \( x = 1.4(y-1.5)^2 - 2.1 \), then plot these points.

Step 6: Sketch the Parabola

Using the vertex and additional points, sketch the parabola. Make sure the curve reflects symmetry about the line \( y = 1.5 \) and opens to the right.

Key Concepts

Standard Form of a Parabola

In mathematics, understanding the standard form of a parabola is crucial for graphing and analyzing these curves. The standard form of a parabola that opens horizontally is given by the equation \[ (y - k)^2 = 4p(x - h) \].Here,

• \((x, y)\) represents any point on the parabola,
• \((h, k)\) is the vertex of the parabola, a significant point that determines the center of symmetry,
• \(4p\) is a constant that affects the "width" of the parabola and the direction it opens.

The equation in the original exercise, \( 1.4(y - 1.5)^2 = 0.5(x + 2.1) \), is in this form. By comparing, we recognize the values:

• \( h = -2.1 \), the \( x \)-coordinate of the vertex,
• \( k = 1.5 \), the \( y \)-coordinate of the vertex,
• and \( 4p = \frac{0.5}{1.4} = \frac{5}{14} \), which helps in determining the parabola's direction.

Identifying these components allows us to easily deduce other properties of the parabola, such as its vertex location and direction of opening.

Vertex of a Parabola

The vertex of a parabola is an essential feature when graphing and understanding the curve. For the equation in standard form \((y-k)^2 = 4p(x-h)\), the vertex is conveniently located at the point \((h, k)\). In the provided exercise, the vertex is \((-2.1, 1.5)\). This point is the "turning point" of the parabola, where the curve changes direction.

• For parabolas that open horizontally, like the one in our exercise, the vertex is where the curve is widest left-to-right.

This central position also serves as a reference for plotting additional points and for ensuring the graph has the correct symmetry about this axis. Additionally, since the vertex's location directly influences where the parabola is on the coordinate plane, it's vital for sketching a precise graph.

Direction of Parabola Opening

The direction in which a parabola opens is determined by the constant \( p \) in the standard form equation. For a horizontally opening parabola \((y-k)^2 = 4p(x-h)\):

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

In our current exercise, we derived that \( p = \frac{5}{56} \). Since this is positive \( (p > 0) \), it indicates the parabola opens towards the positive direction of the x-axis, or rightward.Understanding the direction is key when sketching the parabola, as it affects how we plot additional points around the vertex and interpret solutions graphically. Knowing this, along with the vertex and axis of symmetry, provides a complete framework for accurately crafting the graph on a coordinate plane.