Question

If the graph of an equation is symmetric with respect to the origin and -4 is an \(x\) -intercept of this graph, name another \(x\) -intercept.

Short answer

The other x-intercept is 4.

Step-by-step solution

- Understand Symmetry with Respect to the Origin

If a graph is symmetric with respect to the origin, then for every point \(x, y\), there is a corresponding point \(-x, -y\) on the graph.

- Consider the Given x-Intercept

The given x-intercept is \(-4, 0\). This means that the point \(-4, 0\) lies on the graph of the equation.

- Apply Symmetry to Find the Corresponding Point

Since the graph is symmetric with respect to the origin, the corresponding point to \(-4, 0\) would be \(4, 0\). Therefore, \(4\) is another x-intercept.

Key concepts

origin symmetry

Symmetry is an important concept in graphing equations. When a graph is symmetric with respect to the origin, it means that every point \(x, y\) has a matching point \(-x, -y\). This type of symmetry helps in predicting the behavior and shape of the graph. For example, in the given exercise, if \(-4, 0\) is an x-intercept, then symmetry ensures that \(4, 0\) is also an x-intercept.

x-intercept

An x-intercept is the point where a graph crosses the x-axis. At this point, the y-coordinate is always zero. For instance, if \(-4, 0\) is given as an x-intercept, it means the graph touches the x-axis at this point. Finding additional x-intercepts becomes easier when using properties such as origin symmetry. In this case, it leads to the discovery that \(4, 0\) is also an x-intercept.

coordinate geometry

Coordinate geometry helps us understand the position of points on a plane using pairs of numbers called coordinates. Each point is represented by \(x, y\). When dealing with symmetries, like the origin symmetry, coordinate geometry becomes very useful. It allows us to pinpoint the exact location of corresponding points. The exercise utilizes this to find the x-intercept \(4, 0\) based on the given intercept \(-4, 0\).

algebraic equations

Algebraic equations define relationships between variables. They can be represented graphically. Understanding symmetry and intercepts in the context of these equations provides deeper insights. When a graph shows origin symmetry, the equations can be manipulated to find corresponding points. This principle was used in the exercise to move from the given x-intercept \(-4, 0\) to find \(4, 0\). Knowing that the graph is symmetric with respect to the origin simplified the process.