Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection. $$ \left\\{\begin{array}{c} y^{2}-4 x^{2}=16 \\ y-x=4 \end{array}\right. $$

Step-by-Step Solution

Answer

The graphs intersect at points \( (0, 4) \) and \( \left( \frac{8}{3}, \frac{20}{3} \right) \).

Step 1: Analyze the Equations

The given set of equations is a system of non-linear equations. The first equation, \( y^2 - 4x^2 = 16 \), represents a hyperbola. The second equation, \( y - x = 4 \), is a linear equation that represents a straight line.

Step 2: Solve for One Variable

Start by solving the linear equation for one variable. From \( y - x = 4 \), we can express \( y \) in terms of \( x \): \( y = x + 4 \).

Step 3: Substitute in the First Equation

Substitute \( y = x + 4 \) into the hyperbolic equation. The equation becomes \( (x + 4)^2 - 4x^2 = 16 \).

Step 4: Simplify the Equation

Expand and simplify the equation: \( x^2 + 8x + 16 - 4x^2 = 16 \). This simplifies to \( -3x^2 + 8x = 0 \).

Step 5: Solve the Quadratic Equation

Factor the quadratic equation: \( -x(3x - 8) = 0 \). This gives the solutions \( x = 0 \) and \( 3x - 8 = 0 \), hence \( x = \frac{8}{3} \).

Step 6: Find Corresponding y-values

Substitute the \( x \)-values back into \( y = x + 4 \). For \( x = 0 \), \( y = 4 \). For \( x = \frac{8}{3} \), \( y = \frac{8}{3} + 4 = \frac{20}{3} \).

Step 7: Identify the Points of Intersection

The points of intersection are \( (0, 4) \) and \( \left( \frac{8}{3}, \frac{20}{3} \right) \).

Step 8: Sketch the Graphs

Sketch the hyperbola by plotting points that satisfy the equation \( y^2 - 4x^2 = 16 \). Sketch the line \( y = x + 4 \) by plotting \( y \)-intercept \( (0,4) \) and another point such as \( (1,5) \). Mark the points of intersection on the graph.

Key Concepts

Hyperbola

A hyperbola is a fascinating geometric shape characterized by its distinct curving structure. It is formed by the intersection of a double cone and a plane, where the plane cuts through both halves of the cone. In mathematical terms, a hyperbola is represented by the equation \( y^2 - 4x^2 = 16 \), which is a standard form. This equation describes a set of two mirrored curves that open along the x-axis.

When identifying a hyperbola:

• Observe that the equation has both squared terms, one with a minus sign, indicating the hyperbola's axis of orientation.
• The term \( 4x^2 \) implies that the hyperbola opens horizontally since this term has a negative sign next to it.
• The constant 16 suggests the distance the hyperbola reaches from its center, called the transverse axis.

A hyperbola doesn’t cross axes but rather curves away symmetrically from the center point, making it a unique graphing challenge on a coordinate plane.

Linear Equations

Linear equations, unlike the more complex structures like hyperbolas, represent straight lines on a graph and come in the simplest form of algebraic equations. The equation \( y - x = 4 \) is a perfect example of a linear equation. It can be rearranged into the slope-intercept form \( y = x + 4 \).

Key characteristics of linear equations include:

• The graph is a straight line extending indefinitely in both directions.
• The slope determined by the coefficient of \( x \) indicates the steepness or the inclination of the line relative to the axes. Here, the slope is 1.
• The y-intercept is the starting value of y when \( x = 0 \), which here is 4.

Linear equations are fundamental in algebra, providing a straightforward relationship between variables.

Points of Intersection

When studying graphs, finding the points of intersection is crucial in understanding where two graphs meet. The intersection points of the given equations are \((0, 4)\) and \( \left( \frac{8}{3}, \frac{20}{3} \right) \). These are the x and y coordinates that satisfy both equations simultaneously.

Think of intersection points as shared solutions among system equations:

• They not only serve as visual confirmation that graph calculations are correct but also solve both equations concurrently.
• The points are calculated by substituting variables solved from one equation into the other.
• Graphically, they represent the physical crossing points of the graph lines or curves.

Identifying these intersections aids in deeper analysis of relationships between different graphical elements.

Coordinate Plane

The coordinate plane is an essential aspect of graphing, providing a visual way to describe the relationships between variables. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).

Important features of a coordinate plane include:

• The intersection of the axes is the origin, denoted as \((0,0)\).
• Each point on the plane is represented by a pair of coordinates \((x, y)\), indicating the positions along each axis.
• This system allows graphing of complex shapes, such as hyperbolas, as well as straightforward lines, like linear equations.

Coordinating plane systems help us visualize and solve systems of equations, demonstrating how variables interact in two-dimensional space.

Non-linear Equations

Contrary to linear equations which represent straight lines, non-linear equations can represent various types of curves and more complex geometric shapes. The equation \( y^2 - 4x^2 = 16 \) is non-linear due to the presence of squared terms.

Characteristics of non-linear equations are:

• They can graph shapes like circles, ellipses, parabolas, or hyperbolas depending on their individual structure.
• Unlike linear equations, non-linear ones do not have a constant slope, meaning the graph’s slope changes at every point.
• Solving these equations often requires techniques such as substitution or manipulation to find particular solutions or intersections.

Understanding non-linear equations expands our ability to analyze and predict the behavior of various complex systems. They offer insights beyond simple direct relationships.