Problem 18 - Algebra and Trigonometry

Question

11–18 ? Find the x- and y-intercepts of the graph of the equation. $$ x^{2}-x y+y=1 $$

Step-by-Step Solution

Answer

The x-intercepts are \((1, 0)\) and \((-1, 0)\), and the y-intercept is \((0, 1)\).

Step 1: Finding the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( x^2 - xy + y = 1 \). This simplifies to \( x^2 = 1 \). Solve for \( x \) by finding the square roots: \( x = \pm 1 \). Thus, the x-intercepts are \( (1, 0) \) and \( (-1, 0) \).

Step 2: Finding the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( x^2 - xy + y = 1 \). This results in \( y = 1 \). Therefore, the y-intercept is \( (0, 1) \).

Key Concepts

Understanding the x-intercept

The x-intercept is where a graph crosses the x-axis. This point has a y-coordinate of zero since it's right on the axis. Finding the x-intercept of any equation involves setting \( y = 0 \), then solving the equation for \( x \).

For the quadratic equation \( x^2 - xy + y = 1 \), we set \( y = 0 \) which simplifies the equation to \( x^2 = 1 \). To solve for \( x \), we take the square root of both sides, yielding \( x = \pm 1 \).
These solutions give us the x-intercepts:

• \( (1, 0) \)
• \( (-1, 0) \)

The graph crosses the x-axis at these points, which can be visualized as a part of understanding the overall shape of the graph.

Determining the y-intercept

The y-intercept is where the graph of an equation intersects the y-axis. At this point, the value of \( x \) is zero because it is directly above or below the origin on the graph. To find the y-intercept, we set \( x = 0 \) in the equation and solve for \( y \).

In our equation \( x^2 - xy + y = 1 \), substituting \( x = 0 \) results in the simple equation \( y = 1 \). This gives us the y-intercept:

• \( (0, 1) \)

This point helps in creating a mental image of where the graph is positioned vertically.

What is a quadratic equation?

A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero.

The key feature of quadratic equations is their characteristic U-shaped curve, known as a parabola. When graphing these equations:

• The sign of \( a \) determines the direction of the parabola (upward if positive, downward if negative).
• The vertex is the turning point of the graph.
• The axis of symmetry is a vertical line that passes through the vertex, dividing the graph into two mirror-image halves.

In the given exercise \( x^2 - xy + y = 1 \), while it looks like a quadratic, it behaves a bit differently because of its mixed terms, meaning it may not form a simple parabola, underlining the versatile nature of quadratics.

Graphing quadratic equations

Graphing equations, especially quadratics, starts with understanding their intercepts and shape.

For our specific quadratic, \( x^2 - xy + y = 1 \), plotting the x-intercepts \( (1, 0) \) and \( (-1, 0) \) and the y-intercept \( (0, 1) \) is crucial. These points give a frame for the graph.

The overall shape of a quadratic is a parabola:

• By plotting points and observing symmetry, we can sketch the graph accurately.
• The intercepts help locate the curve on the plane, giving us crucial insight into its behavior.
• Given additional points or using graphing tools helps ensure precision in more complex forms.

Understanding these elements allows for effective graphing of quadratic equations, just like reading a map to understand a landscape.