Problem 17 - Algebra and Trigonometry

Question

11–18 ? Find the x- and y-intercepts of the graph of the equation. $$ x y=5 $$

Step-by-Step Solution

Answer

The graph of \( xy = 5 \) has no x-intercept or y-intercept.

Step 1: Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( xy = 5 \). Substitute \( y = 0 \) and solve for \( x \):\[ x \cdot 0 = 5 \]\[ 0 = 5 \]This equation is not possible, indicating there is no x-intercept.

Step 2: Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( xy = 5 \). Substitute \( x = 0 \) and solve for \( y \):\[ 0 \cdot y = 5 \]\[ 0 = 5 \]This equation is also not possible, indicating there is no y-intercept.

Step 3: Interpret Results

Since neither the x-intercept nor the y-intercept can be found, the graph of the equation \( xy = 5 \) does not intersect the x-axis or y-axis. This equation represents a hyperbola, which does not cross the axes in this standard form.

Key Concepts

Understanding x-intercepts

An x-intercept is where a graph crosses the x-axis. This means that at this point, the value of y is zero. For most algebraic equations, finding the x-intercept involves setting y equal to zero and solving for x.

• For example, in the equation of a straight line, you substitute y = 0 and solve for x.
• This will give you the point on the x-axis where the line crosses it.

In the case of our equation, \( xy = 5 \), setting \( y = 0 \) leads to a contradiction. The equation simplifies to \( x \times 0 = 5 \), which is impossible since zero times anything is zero, not five. Therefore, the graph of this equation has no x-intercept. This result suggests that the graph never crosses the x-axis, which is a special characteristic found in hyperbolas.

Understanding y-intercepts

A y-intercept is where a graph crosses the y-axis. Here, the x-value is zero, meaning we set x = 0 and solve for y in algebraic equations.

• For a linear equation, this typically yields a numerical value which is the point at which the line or curve crosses the y-axis.
• For example, the equation \( y = mx + b \) gives the y-intercept as \( b \) when \( x = 0 \).

In our situation, when we substitute \( x = 0 \) into \( xy = 5 \), the equation becomes \( 0 \times y = 5 \), again leading to a contradiction, implying no y-intercept exists. This is similar to what we observed with the x-intercept, reinforcing that specific graphs such as hyperbolas behave differently from lines.

Introduction to Hyperbolas in Algebra

A hyperbola is a type of curve on a graph, characterized by its two separate branches. These curves often do not intersect the axes depending on their orientation and structure.

• Hyperbolas arise in equations where the product of variables is constant, like \( xy = k \).
• This creates two symmetric, open curves that move towards infinity in opposite quadrants of the graph.

In the case of our algebraic equation \( xy = 5 \), the hyperbola forms in the standard position where neither of its branches crosses the x- or y-axis, as seen by the absence of intercepts. Understanding the distinct nature of hyperbolas in algebra can help predict graph behaviors without even plotting them.

Algebraic Equations and Their Graphical Representations

Algebraic equations form the foundation for many types of graphs, including lines, circles, parabolas, and hyperbolas. Each type has unique properties, especially considering how they might or might not intersect axes.

• An equation like \( y = mx + b \) represents a straight line which nearly always crosses the x and y-axes unless it's parallel to them.
• Circular and parabolic equations typically intersect axes but can miss them entirely under specific conditions.

The equation \( xy = 5 \) is a hyperbola, representing an entirely different kind of behavior: none of the axes intersect it. By exploring these aspects, students gain insights into the behavior of different algebraic expressions when graphed, bridging between algebra and geometry in practical ways.