Problem 48 - Algebra and Trigonometry

Question

\(41-48\) 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: Understand the x-intercept

To find the x-intercept, set the y-value to 0 in the equation and solve for x. This is because at the x-intercept, the value of y is 0.

Step 2: Solve for x-intercept

Substitute \(y = 0\) into the equation \(x^2 - xy + y = 1\):\[x^2 - x(0) + 0 = 1\]Which simplifies to:\[x^2 = 1\]Take the square root of both sides to solve for x:\[x = \pm 1\]Therefore, the x-intercepts are \((1, 0)\) and \((-1, 0)\).

Step 3: Understand the y-intercept

To find the y-intercept, set the x-value to 0 in the equation and solve for y. This is because at the y-intercept, the value of x is 0.

Step 4: Solve for y-intercept

Substitute \(x = 0\) into the equation \(x^2 - xy + y = 1\):\[(0)^2 - 0(y) + y = 1\]Which simplifies to:\[y = 1\]So, the y-intercept is \((0, 1)\).

Key Concepts

x-intercept

The x-intercept is the point where a graph intersects the x-axis. At this point, the y-value is always zero. To find the x-intercept of the equation \(x^2 - xy + y = 1\), we set \(y\) to 0 in the equation. The simplified equation becomes \(x^2 = 1\). Solving \(x^2 = 1\) involves taking the square root of both sides, resulting in two possible solutions: \(x = 1\) and \(x = -1\). This means there are two x-intercepts for the given equation.- When \(x = 1\), the point is \((1, 0)\).- When \(x = -1\), the point is \((-1, 0)\).
Thus, the graph intersects the x-axis at these two points.

y-intercept

The y-intercept refers to the point where a graph intersects the y-axis. At this intersection, the x-value is zero. To calculate the y-intercept for the equation \(x^2 - xy + y = 1\), we substitute \(x = 0\) into the equation. This changes the equation to \(y = 1\), which is simple to solve. The solution \(y = 1\) provides us with the coordinates of the y-intercept.The point where the graph crosses the y-axis is \((0, 1)\).
It's the single y-intercept for this quadratic equation. Understanding this point helps in sketching and understanding the position of the curve in relation to axes.

graphing equations

Graphing equations involves plotting points that satisfy the equation onto a coordinate grid. The specific shape of the graph for a quadratic equation often forms a parabola. For the equation \(x^2 - xy + y = 1\), identifying intercepts provides key points that guide the drawing.

• The x-intercepts \((1, 0)\) and \((-1, 0)\) tell us where the graph crosses the x-axis.
• The y-intercept \((0, 1)\) shows where it crosses the y-axis.

By plotting these points and understanding the quadratic nature of the equation, you can draw a curve that represents the equation's solutions.It's essential to consider the symmetry typical of quadratics, usually about a vertical line (the axis of symmetry).
Knowing intercepts simplifies this process by providing fixed points to anchor the curve, ensuring more accurate graphing.