Problem 64 - Contemporary Precalculus

Question

Use the standard viewing window to graph the function f and the function \(g(x)=|f(x)|\) on the same screen. Exercise 66 may be helpful for interpreting the results. $$f(x)=x+3$$

Step-by-Step Solution

Answer

Answer: The graph of g(x) is a reflection of the graph of f(x) onto the x-axis for all negative values of f(x). The positive values of f(x) remain the same. In other words, g(x) is the absolute value of f(x), resulting in a V-shaped graph with the vertex at (-3,0).

Step 1: Identify the equation for g(x)

As we know, g(x) is the absolute value of f(x). The given function f(x) is \(f(x)=x+3\). So, we will substitute f(x) into g(x) definition to get the equation for g(x): $$g(x)=|f(x)|=|x+3|$$

Step 2: Find the key points

For both functions, we will identify a few key points to help us in graphing them. For f(x), we can easily find the x and y-intercepts: $$f(0)=3 \Rightarrow \textrm{y-intercept at }(0,3)$$ $$x+3=0 \Rightarrow x=-3 \Rightarrow \textrm{x-intercept at }(-3,0)$$ For g(x), we will find the minimum point of the absolute value function: $$x+3=0 \Rightarrow x=-3 \Rightarrow g(x)=|-3+3|=0 \Rightarrow \textrm{Minimum point at}(-3,0)$$

Step 3: Plot the key points and draw the graph

Now plot the key points for both functions onto the same set of axes. For the function f(x): 1. Plot y-intercept at (0,3) 2. Plot x-intercept at (-3,0) 3. Draw a straight line that goes through these two points. For the function g(x): 1. Plot the minimum point at (-3,0) 2. Since g(x) is an absolute value function, its graph will be V-shaped. The vertex of the V will be at the minimum point we found. 3. Draw the V-shaped graph with the two linear parts on either side of the vertex. Now, we have the graphs of both f(x) and g(x) on the same screen. The graph of f(x) is a straight line with a slope of 1, while the graph of g(x) is a V-shaped graph with its vertex at (-3,0), opening upwards.

Key Concepts

Absolute Value Functions

The concept of absolute value in mathematics represents the distance of a number from zero on the number line, regardless of direction. An absolute value function is written as \(g(x) = |f(x)|\), where \(f(x)\) is another function.

An essential feature of absolute value functions is their characteristic V-shaped graph. This shape occurs because absolute values are always non-negative—when plotted, this results in a graph that sharply turns at the point where \(f(x)\) equals zero, known as the vertex. The absolute value functions are mirrored across the x-axis at this vertex point, creating the 'V' shape.

When graphing \(g(x) = |f(x)|\), such as \(g(x) = |x+3|\) from the exercise, it's crucial to determine the vertex. This is done by setting \(f(x)\) to zero and solving for \(x\), as seen with \(x = -3\) in the example. The vertex is at the point where the 'inside' of the absolute value, \(x+3\), reaches zero turning the entire expression \(g(x)\) to zero as well, thus creating the minimum point at (-3,0) for the V-shape.

Linear Functions

Linear functions, which form one of the most fundamental types of functions, are algebraic equations that produce a straight line when graphed. The standard form is \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, the point where the line crosses the y-axis.

The slope \(m\) illustrates how steep the line is; a larger absolute value of \(m\) means a steeper line. The y-intercept \(b\) gives a starting point for drawing the line on a graph. In our example, the linear function is \(f(x) = x + 3\), indicating a slope of 1 and a y-intercept at (0,3).

To graph a linear function, you need only two points. Often, the x- and y-intercepts are the easiest to calculate and use. Once these points are plotted on a coordinate plane, you simply draw a line through them to represent the function. It’s the simplicity of linear functions that often makes them the starting point for understanding more complex functions.

Intercepts

Intercepts on a graph are where the function crosses the axes of the coordinate plane. There are two types of intercepts: x-intercepts and y-intercepts. The x-intercept occurs where the function’s graph crosses the x-axis, which means the y-value is zero at this point. Conversely, the y-intercept occurs where the function’s graph crosses the y-axis, so the x-value is zero.

Identifying the intercepts usually involves setting \(x\) to zero to find the y-intercept, and setting the function itself to zero to find the x-intercept. In the context of our exercise, the linear function \(f(x) = x + 3\) has a y-intercept at (0, 3) because when \(x\) is zero, \(f(x)\) equals 3. Similarly, the x-intercept is found by setting \(f(x)\) to zero, resulting in the equation \(x + 3 = 0\), which solves to \(x = -3\), giving an x-intercept at (-3, 0).

Intercepts are particularly useful when graphing functions, as they provide clear points to start the graph, and in the case of absolute value and linear functions, they can often determine the shape and direction of the graph entirely.