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Chapter 1: Problem 31

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(y^{2}=6-x\)

Short Answer

Expert verified
The \(x\)-intercept is \(x=6\), the \(y\)-intercepts are \(y=\sqrt{6}\) and \(y=-\sqrt{6}\).

Step by step solution

01

- Find the x-intercept

To find the \(x\)-intercept, set \(y = 0\) in the equation so we have \(0^{2} = 6-x \rightarrow x = 6\).
02

- Find the y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the equation which gives \(y^{2} = 6\). This yields two solutions, \(y = \sqrt{6}\) and \(y=-\sqrt{6}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the x-intercept
To find the x-intercept of a graph, you need to determine where the graph crosses the x-axis. For this to occur, the y-value must be zero because any point on the x-axis has a y-value of zero. So, you simply substitute 0 into the equation for y and solve for x.
In the provided equation \(y^{2}=6-x\), by setting \(y=0\), the equation simplifies to \(0^{2}=6-x\), which leads to \(x=6\).
This tells us that the x-intercept of the graph is at the point \((6,0)\).
  • Substitute \(y=0\) to find x-intercept.
  • Simplify the equation to solve for \(x\).
  • The x-intercept is where the graph touches the x-axis.
Understanding x-intercepts is crucial as they reveal when an expression equals zero, helping in various calculations and graph interpretations.
Exploring the y-intercept
The y-intercept of a graph is where the graph crosses the y-axis. At this point, the x-value must be zero because any point on the y-axis has a zero x-value.
For the equation \(y^{2}=6-x\), set \(x=0\) to find the y-intercept. Substituting into the equation gives \(y^{2}=6\).
Solving for y gives us two solutions: \(y=\sqrt{6}\) and \(y=-\sqrt{6}\). This means there are two y-intercepts at \((0, \sqrt{6})\) and \((0, -\sqrt{6})\).
  • Set \(x=0\) to find the y-intercept.
  • Solve the equation for \(y\).
  • Graphs can intersect the y-axis at more than one point, leading to multiple y-intercepts.
The y-intercept provides insight into where the graph intersects the y-axis, offering valuable information about the graph's behavior.
Solving quadratic equations
Solving quadratic equations involves finding the value(s) of the variable that make the equation true. In the context of finding intercepts, solving \(y^2=6-x\) meant setting it up as a quadratic in x or y.
Quadratic equations typically have the form \(ax^2 + bx + c = 0\). The methods to solve them include:
  • Factoring: Splitting the equation into two binomials.
  • Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).
  • Completing the square: Rewriting the equation to make it easier to solve.
  • Graphical method: Using a graph to find where the curve intersects the axes.
For the y-intercept in our example, we found \(y\) by solving the equation \(y^2=6\), which resulted in two potential values for \(y\).
Each method may be more suitable depending on the equation you are tackling. Understanding these solutions helps unravel the roots or intercepts that are integral to the equation's graph.

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