Problem 73 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=|x-4|\)

Step-by-Step Solution

Answer

The graph of the equation \(y=|x-4|\) is a V-shape that intersects the x-axis at (4,0) and does not intersect the y-axis. It also shows no symmetry about the y-axis.

Step 1: Graphing the Function

The graph of the function \(y=|x-4|\) is a V-shaped graph that opens upwards, with its vertex at (4,0). The value of x in the function \(|x-4|\) is subtracted from 4, causing the graph to shift to the right on the x-axis compared to the basic absolute value function \(y=|x|\). The graph is a straight line to the right of (4, 0) with a slope of +1, and a straight line to the left of (4, 0) with a slope of -1.

Step 2: Finding Intercept Points

The x-intercept for this graph will occur when \(y=0\), which happens when \(x=4\), hence the x-intercept is (4,0). For the function \(y=|x-4|\), it will only cross the x-axis at the point (4,0), hence the y-intercept does not exist for this function.

Step 3: Testing for Symmetry

To test for symmetry, replace \(x\) with \(-x\) in the equation and simplify. If the resulting equation is identical to the original equation, then the graph is symmetric about the y-axis. If not, then it's not symmetric. For the given function \(y=|x-4|\), when we replace \(x\) with \(-x\), we get \(y=|-x-4|\) which is not same as the original equation. So, the graph of the function \(y=|x-4|\) is not symmetric about the y-axis.

Key Concepts

Understanding Absolute Value Transformations

When we talk about graphing absolute value functions, understanding transformations is key. The absolute value function typically has a 'V' shape and is expressed as f(x)=|x|. However, transformations can shift, reflect, or stretch this graph. In our exercise, the function y=|x-4| is a transformation of the basic absolute value function.

This particular transformation involves a horizontal translation. The -4 inside the absolute value indicates that the entire graph of f(x)=|x| has been shifted to the right by 4 units. This moves the vertex of the V from the origin (0,0) to (4,0). These shifts do not affect the shape of the graph but rather its position in the coordinate plane.

Transformations can include reflections over the x or y-axis and vertical stretches or compressions determined by a coefficient in front of the absolute value. It's crucial to understand how each type of transformation affects the graph so that we can accurately depict the functions visually.

Deciphering the X-Intercept in Graphs

An x-intercept is a point where a graph crosses the x-axis. Finding the x-intercept is a fundamental aspect when sketching the graph of an equation, as it gives us a point of reference. In the context of our exercise, the function y=|x-4| has its x-intercept at the point where the graph crosses the x-axis, which occurs when y=0.

To find the x-intercept, we set y to zero and solve for x. Doing this for our function yields the equation 0=|x-4|. The solution to this equation is x=4, hence the graph will have an x-intercept at (4,0). This point also acts as the vertex of the V-shaped graph. When dealing with absolute value functions, there can be more than one x-intercept; however, for this particular function, there is only one. This is important when sketching the graph, as every x-intercept represents a crucial touchpoint for the curve.

Exploring Symmetry in Graphs

Symmetry in graphs can greatly simplify the understanding and drawing of functions. A graph can be symmetric about the y-axis, the x-axis, or the origin. For absolute value functions, it's common to see symmetry about the y-axis due to their V-shaped nature. However, this is only true for the basic function f(x)=|x|.

To investigate the symmetry of the function y=|x-4| from our exercise, we attempt a symmetry test. We replace each x with -x and observe if the modified function is equivalent to the original. This exercise's function, when substituted with -x, leads to y=|-(-x)-4| or y=|x+4|, which is not equivalent to the original function. Consequently, the graph lacks y-axis symmetry, displaying how horizontal shifts affect the symmetry properties of the function. Recognizing these properties is vital for accurately analyzing and sketching the behavior of the function on the graph.