Question

If the line given by the equation \(y=4 x+7\) is reflected about the \(x\) -axis, what will be the graph of the resulting function?

Short answer

The reflected line is given by the equation \\(y = -4x - 7\\).

Step-by-step solution

Understanding Reflection Over the x-axis

Reflecting a graph over the x-axis means that each point on the graph is mirrored across the x-axis. This transforms the y-coordinate of each point to its opposite (or negative), while the x-coordinate remains unchanged. Thus, for a given point \(x, y\), it becomes \(x, -y\).

Original Line Equation Inspection

The line equation given is \(y = 4x + 7\). This is a linear equation in slope-intercept form \(y = mx + b\), where the slope \(m\) is 4 and the y-intercept \(b\) is 7. Each point \(x, y\) on this line satisfies this equation.

Applying Reflection to the Line Equation

To reflect the line over the x-axis, replace \(y\) with \(-y\). Thus the equation becomes \(-y = 4x + 7\). This represents the points after reflection.

Rearranging the Reflective Equation

Rearrange \(-y = 4x + 7\) to express \(y\) by multiplying the entire equation by \(-1\), yielding \(y = -4x - 7\). This is the equation of the new line after reflection, presented in the slope-intercept form.

Key concepts

Reflection Over the X-axis

Reflecting a graph over the x-axis involves flipping all points on the graph to form a mirror image across the x-axis. In simple terms, each point on the graph has its y-coordinate changed to its opposite (or negative), while the x-coordinate remains unchanged.
For example, the point \(3, 5\) on a graph will become \(3, -5\) after reflection.
This concept is crucial to understand when dealing with graph transformations, as it alters the graph's appearance without moving it horizontally. The resulting graph maintains the same shape but is positioned inverted relative to the x-axis.

Linear Equation

A linear equation is a mathematical expression that forms a straight line when graphed in a coordinate plane. It typically takes the form \(y = mx + b\), where \(m\) and \(b\) are constants. Here, \(m\) represents the slope of the line, and \(b\) represents the y-intercept, which is the point where the line crosses the y-axis.
Key features of a linear equation include:

• Straight-line graph.
• Constant slope (uniform rate of change).
• Y-intercept (where the line crosses the y-axis).

Understanding linear equations allows you to predict how a line will behave on a graph, whether through changes in slope or shifts using transformations like reflection.

Slope-intercept Form

The slope-intercept form of a linear equation is one of the most commonly used forms to express linear equations. Written as \(y = mx + b\), it provides clear information about the slope and y-intercept at a glance.
Slope \(m\): This value indicates the steepness and direction of the line. Positive slopes rise as they move from left to right, whereas negative slopes fall.
Y-intercept \(b\): This constant tells you where the line intersects the y-axis. It’s a crucial starting point for sketching your line on the graph.
By having linear equations in slope-intercept form, you can easily graph functions and translate transformations like reflections efficiently.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures through a coordinate system. By using coordinates, specifically \(x\) and \(y\), we can describe the position and relationship of points, lines, and shapes in two-dimensional space.
Important components of coordinate geometry include:

• Coordinate System: A two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis).
• Points: Identified by coordinates (x, y) that determine their position on the plane.
• Lines and Curves: Described using equations that relate x and y, like linear equations.

Understanding coordinate geometry is vital for solving problems involving graphical representations and transformations, such as reflections and translations, making it a foundational topic in mathematics.