Question

If the graph of an equation is symmetric with respect to the \(y\) -axis and 6 is an \(x\) -intercept of this graph, name another \(x\) -intercept.

Short answer

The other x-intercept is -6.

Step-by-step solution

- Understand the symmetry

If a graph is symmetric with respect to the y-axis, then for every point \(x, y\) on the graph, there is a corresponding point \(-x, y\). This means the graph mirrors itself on the left and right sides of the y-axis.

- Identify the given x-intercept

The given x-intercept is 6. This means that there is a point (6, 0) on the graph.

- Use symmetry to find another x-intercept

Since the graph is symmetric with respect to the y-axis, the corresponding point to (6, 0) is (-6, 0). Therefore, -6 is another x-intercept.

Key concepts

y-axis symmetry

When we talk about a graph being symmetric with respect to the y-axis, it means that the left and right sides of the graph are mirror images of each other. For any point \(x, y\) on the graph, there exists a corresponding point \(-x, y\). This kind of symmetry makes graphs visually balanced and easier to analyze. \(x\)-values on the positive and negative sides of the y-axis will lead to the same \(y\)-value.
For example, if a graph includes the point \( (3, 2) \), then it must also include \( (-3, 2) \) if it is symmetric with respect to the y-axis. This mirroring helps us predict and find corresponding points more effortlessly.
Understanding this property of y-axis symmetry is crucial when solving problems that require identifying other intercepts or points.

x-intercepts

The x-intercepts of a graph are the points where the graph crosses the x-axis. At these points, the value of y is zero. To find x-intercepts, we typically set \(y = 0\) in the equation of the graph and solve for \( x \).
For a graph symmetric about the y-axis, knowing one x-intercept can help you find another. If \( 6 \) is given as an x-intercept (meaning the graph crosses at \( (6,0) \)), then due to the symmetry, \(-6 \) will also be an x-intercept (or \( (-6,0) \)).
The simplicity of finding such symmetrical intercepts can save time and make solving graph-related problems easier. Whenever you are given an x-intercept and know the graph's symmetry properties, you can quickly determine additional intercepts.

graphical analysis

Graphical analysis involves visually interpreting the characteristics and behavior of a graph, such as intercepts, symmetry, and shape. By examining a graph, we can gain insights into the corresponding algebraic equation and the behavior of the function it represents.
When faced with a graph that is symmetric about the y-axis, one can efficiently analyze the intercepts and other key features. For instance, recognizing that such a graph will have mirrored points can reveal hidden characteristics and simplify problem-solving.

• Determine the x-intercepts quickly.
• Observe the overall shape and structure of the graph.
• Compare symmetrical points to verify calculations.

Understanding these elements will enhance your ability to solve graph-related problems accurately and swiftly. Visual tools complement algebraic methods, offering a comprehensive approach to analyzing mathematical functions.