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Chapter 2: Problem 62

Find the \(x\) - and \(y\) -intercepts. $$ x=y^{2}-4 $$

Short Answer

Expert verified
The x-intercept is (-4, 0) and the y-intercepts are (0, 2) and (0, -2).

Step by step solution

01

Find the y-intercept

To find the y-intercept, set the value of x to 0 in the equation. This gives: \[\begin{equation} 0 = y^2 - 4 onumber \ y^2 - 4 = 0 onumber \ y^2 = 4 onumber \ y = \boxed{\text{\textpm}2} onumber \end{equation}\] Therefore, the y-intercepts are at (0, 2)and (0, -2).
02

Find the x-intercept

To find the x-intercept, set the value of y to 0 in the equation. This gives: \[\begin{equation} x = 0^2 - 4 onumber \ x = -4 onumber \end{equation}\] Therefore, the x-intercept is at (-4, 0).

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Key Concepts

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

x-intercept
The x-intercept is where a graph crosses the x-axis. To find it, set the value of y to 0 in the equation. For the equation \(x = y^2 - 4\), substitute y with 0. This simplifies to \(x = 0^2 - 4\), resulting in \(x = -4\). Therefore, the point (-4, 0) represents the x-intercept. Interpreting this visually, the x-intercept shows where the graph touches or crosses the x-axis.
y-intercept
The y-intercept is where a graph crosses the y-axis. To find it, set the value of x to 0 in the equation. For the equation \(x = y^2 - 4\), set x to 0, leading to \(0 = y^2 - 4\). Solving for y, we add 4 to both sides to get \(y^2 = 4\). Taking the square root of both sides, we find \(y = \textpm 2\). This means the y-intercepts are at (0, 2) and (0, -2) where the graph crosses the y-axis. These points indicate where the graph intersects the y-axis.
quadratic equations
Quadratic equations form parabolas that may open upwards, downwards, or sideways. They take the form \(ax^2 + bx + c = 0\) or similar. In our case, we have \(x = y^2 - 4\). Solving quadratic equations typically involves methods like factoring, completing the square, or using the quadratic formula. For our equation, it describes a sideways parabola. Understanding the type and shape of the parabola helps in visualizing the intercepts and graph.
problem-solving steps
Step-by-step problem-solving is essential in algebra. Here's how it applies to finding intercepts:
  • Find y-intercept: Set x to 0, solve for y. For \(x = y^2 - 4\), setting x to 0 yields \(0 = y^2 - 4\), and solving it gives y as \(2\) and \(-2\).
  • Find x-intercept: Set y to 0, solve for x. In our equation, setting y to 0 gives \(x = -4\).
Following these steps helps organize the solution process and ensures accuracy when finding intercepts.

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