Question

The solar electric generating systems at Kramer Junction, California, use parabolic troughs to heat a heat-transfer fluid to a high temperature. This fluid is used to generate steam that drives a power conversion system to produce electricity. For troughs 7.5 feet wide, an equation for the cross section is \(16 y^{2}=120 x-225\). (a) Find the intercepts of the graph of the equation. (b) Test for symmetry with respect to the \(x\) -axis, the \(y\) -axis, and the origin.

Short answer

x-intercept: (15/8, 0). No y-intercept. Symmetric about the x-axis, not the y-axis or origin.

Step-by-step solution

- Rewrite the equation

Rewrite the equation in standard form, which is more recognizable for intercepts and symmetry analysis. The given equation is: \[ 16y^2 = 120x - 225 \].Divide every term by 16 to simplify it:\[ y^2 = \frac{120x - 225}{16} \].

- Find x-intercept

To find the x-intercept, set \( y = 0 \) in the equation and solve for \( x \):\[ 0 = \frac{120x - 225}{16} \].Since \( 0 = 120x - 225 \), solving for \( x \) yields:\[ 120x = 225 \].Thus, \( x = \frac{225}{120} = \frac{15}{8} \). So the x-intercept is \( \left( \frac{15}{8}, 0 \right) \).

- Find y-intercept

To find the y-intercept, set \( x = 0 \) in the equation and solve for \( y \):\[ y^2 = \frac{120(0) - 225}{16} \].This simplifies to:\[ y^2 = -\frac{225}{16} \].Since \( y^2 \) cannot be negative, there is no y-intercept.

- Test for symmetry with respect to the x-axis

Replace \( y \) with \( -y \) in the equation and check if the equation remains the same:\[ 16(-y)^2 = 120x - 225 \].Since \( 16y^2 = 120x - 225 \), it remains unchanged. Therefore, the graph is symmetric with respect to the x-axis.

- Test for symmetry with respect to the y-axis

Replace \( x \) with \( -x \) in the equation and check if the equation remains the same:\[ 16y^2 = 120(-x) - 225 \].This changes the term to \( -120x - 225 \), which is not the original equation. Therefore, the graph is not symmetric with respect to the y-axis.

- Test for symmetry with respect to the origin

Replace \( x \) with \( -x \) and \( y \) with \( -y \) in the equation and check if it remains the same:\[ 16(-y)^2 = 120(-x) - 225 \].This simplifies to \[ 16y^2 = -120x - 225 \], which is not the original equation. Therefore, the graph is not symmetric with respect to the origin.

- Summarize findings

The x-intercept is \( \left( \frac{15}{8}, 0 \right) \). There is no y-intercept. The graph is symmetric with respect to the x-axis but not with respect to the y-axis or the origin.

Key concepts

Intercepts

Intercepts are the points where a graph crosses the axes. They provide important clues about the behavior of the graph. There are two types of intercepts: x-intercepts and y-intercepts.

• **X-intercept:** Found by setting \(y = 0\) in the equation and solving for \(x\).
• **Y-intercept:** Found by setting \(x = 0\) in the equation and solving for \(y\).

For the given equation, \[16 y^{2} = 120 x - 225\], let's derive the intercepts:

*X-Intercept*:
Setting \(y = 0\), we get:
\[0 = \frac{120x - 225}{16}\]
This simplifies to:
\[120x = 225\]
So, the x-intercept is \( \frac{15}{8}\).

*Y-Intercept*:
Setting \(x = 0\), we get:
\[y^{2} = \frac{-225}{16} = -\frac{225}{16}\]
Since \(y^{2}\) cannot be negative, there is no y-intercept.

Symmetry

Symmetry in graphs helps to understand their balance and repetitive pattern. In coordinate geometry, we commonly examine symmetry with respect to the x-axis, y-axis, and origin.

• **X-axis Symmetry:** Replace \(y\) with \(-y\) and check if the equation remains unchanged.
• **Y-axis Symmetry:** Replace \(x\) with \(-x\) and check if the equation remains unchanged.
• **Origin Symmetry:** Replace \(x\) with \(-x\) and \(y\) with \(-y\), then check if the equation stays the same.

For the given equation \[16 y^{2} = 120x - 225\]:

*X-axis Symmetry:*
Replace \(y\) with \(-y\):
\[16(-y)^{2} = 120x - 225\]
Since \[16 y^{2} = 120x - 225\], the graph is symmetric about the x-axis.

*Y-axis Symmetry:*
Replace \(x\) with \(-x\):
\[16 y^{2} = 120(-x) - 225\]
\[16 y^{2} = -120x - 225\] is not the original equation, so no y-axis symmetry.

*Origin Symmetry:*
Replace \(x\) with \(-x\) and \(y\) with \(-y\):
\[16(-y)^{2} = 120(-x) - 225\]
\[16 y^{2} = -120x - 225\] is not the original equation, so no symmetry with respect to the origin.

Parabolic Equations

Parabolic equations describe the graphs of parabolas, which are U-shaped curves. They can open upwards, downwards, left, or right.

A common form is \[y = ax^{2} + bx + c\], but other forms like \[Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0\] also exist.

For the given problem, the equation \[16 y^{2} = 120 x - 225\] represents a horizontal parabola opening rightwards. Here's why:

• The equation is rewritten as \[y^{2} = \frac{120x - 225}{16}\] showing \(y^{2}\) in terms of \(x\), typical for parabolas that open horizontally.
• If \(y^{2}\) is isolated, positive \(x\) shifts the vertex to the right, indicating the opening direction.

Understanding the structure helps to visualize and analyze the graph's properties and behavior.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using algebra to study geometric properties and spatial relationships. Points are plotted using coordinates (x, y) on a plane.

Key concepts include:

• **Distance Formula:** Measures the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) given by \[ \text{Distance} = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}\]
• **Midpoint Formula:** Finds the midpoint of a line segment joining \((x_1, y_1)\) and \((x_2, y_2)\) given by \[ \text{Midpoint} = \left( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right) \]

Using coordinate geometry helps dissect the problem effectively:
* The distance from the vertex to the focus in a parabola highlights its shape.
* Midpoints help identify central points in symmetrical objects.
Combining these principles allows deeper insight into the parabolic equations and their respective graphs.