Problem 47 - Precalculus: Functions and Graphs

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. $$\left\\{\begin{aligned}y^{2}-4 x^{2} &=16 \\\y-x &=4\end{aligned}\right.$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Equations

We have two equations here: the first equation \(y^2 - 4x^2 = 16\) represents a hyperbola, and the second equation \(y - x = 4\) represents a straight line.

Step 2: Solve for y in the Linear Equation

Isolate \(y\) in the linear equation: \(y - x = 4\). This gives us \(y = x + 4\).

Step 3: Substitute in the Hyperbola Equation

Substitute \(y = x + 4\) from the linear equation into the hyperbola equation: \((x+4)^2 - 4x^2 = 16\).

Step 4: Simplify and Solve the Quadratic Equation

Expand and simplify the equation:\[(x+4)^2 - 4x^2 = x^2 + 8x + 16 - 4x^2 = 16\].This simplifies to \(-3x^2 + 8x + 16 = 16\). Subtract 16 from both sides: \[-3x^2 + 8x = 0\]. Factor out \(x\): \[-x(3x - 8) = 0\].Solve for \(x\): \(x = 0\) or \(x = \frac{8}{3}\).

Step 5: Find Corresponding y-Values

Substitute \(x = 0\) into the equation \(y = x + 4\): - When \(x = 0\), \(y = 0 + 4 = 4\). Substitute \(x = \frac{8}{3}\) into \(y = x + 4\): - When \(x = \frac{8}{3}\), \(y = \frac{8}{3} + 4 = \frac{20}{3}\).

Step 6: Identify Points of Intersection

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

Step 7: Sketch the Graphs

Draw the hyperbola \(y^2 - 4x^2 = 16\), which opens up and down, and the line \(y = x + 4\). Mark the intersection points \((0, 4)\) and \(\left(\frac{8}{3}, \frac{20}{3}\right)\) on the graph.

Key Concepts

Hyperbola

A hyperbola is a fascinating curve that can appear quite complex at first glance. Unlike circles and ellipses, a hyperbola opens outwards and consists of two separate branches. This divergence occurs because a hyperbola is defined by a difference of squares.

In mathematical terms, one common form of a hyperbola is given by the equation \[ y^2 - 4x^2 = 16 \].This equation tells us that the hyperbola will open vertically since the \(y^2\) term is positive. The constant on the right side (16 in this case) dictates how wide the branches are apart.

Hyperbolas have two axes:

• The transverse axis, which joins the two vertices of the branches.
• The conjugate axis, which runs perpendicular to the transverse axis.

Understanding these features helps in sketching and visualizing hyperbolas effectively.

Linear Equation

Linear equations represent straight lines in coordinate geometry. These are the simplest types of equations and have a constant rate of change. In our example, we have the equation \( y - x = 4 \),which can be rewritten as \( y = x + 4 \)for easier interpretation.

This form of the equation \( y = mx + c \) is known as the slope-intercept form, where \( m \) is the slope and \( c \) is the y-intercept.

• The slope \( m \) measures the line's steepness. A slope of 1 means the line moves diagonally upward at a 45-degree angle.
• The y-intercept \( c \) is where the line crosses the y-axis. Here, it crosses at \( (0, 4) \).

Linear equations are easy to graph, making them helpful as reference points when plotting additional geometric shapes like curves or polygons.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebra to understand geometric concepts. It allows you to solve problems by placing figures like lines and curves in a coordinate plane for better visual comprehension.

In this exercise, the intersection of the hyperbola \( y^2 - 4x^2 = 16 \)and the line \( y - x = 4 \)is found using coordinate geometry. By solving these equations simultaneously, you determine the points at which the line and hyperbola intersect.

• This involves isolating one variable, substituting it into the other equation, and solving the resulting equation.
• The intersection points give you solutions, which are sometimes real, complex, or nonexistent depending on the equations involved.

Mastering coordinate geometry equips you with essential skills for studying higher-dimensional geometry.

Graphing Equations

Graphing equations is an essential skill that visually represents algebraic equations on a plane. It bridges the gap between abstract mathematical expressions and their visual representations.

Let's consider graphing the hyperbola and the line together on a coordinate plane:
Start by identifying key characteristics:

• For the hyperbola \( y^2 - 4x^2 = 16 \),identify the center and the direction in which it opens. The plot will consist of two branches opening vertically.
• For the line \( y = x + 4 \),plot the y-intercept at \( (0, 4) \) and use the slope to identify another point.

Next, provide a rough sketch showing where the hyperbola and line potentially intersect. These are visually estimated points that help confirm the calculations. Even though graphing may seem tedious at first, with practice, it becomes a powerful tool for approaching and solving complex algebraic problems.