Problem 81 - College Algebra Essentials

Question

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$g(x)=|x|+4$$

Step-by-Step Solution

Answer

The square root function \(f(x)=\sqrt{x}\) is half a parabolic shaped graph over the positive x-axis that starts from the origin. The absolute value function \(g(x)=|x|+4\) is a V-shaped graph that starts from the point (0,4) and is defined for all real numbers.

Step 1: Graph the square root function

Begin by placing some values for \(x\) and corresponding \(y = \sqrt{x}\) on the coordinate plane. Set \(x=0;1;4;9\) etc., because they all have integer square roots, hence \(y = 0;1;2;3\) respectively. The points will be \((0,0); (1,1); (4,2); (9,3)\) etc. Because \(f(x)=\sqrt{x}\) is only defined for \(x\geq0\), plot these points and sketch the graph for positive \(x\) values. The graph is typically a curve starting from the origin (0,0) and extending upwards and to the right.

Step 2: Graph the absolute function

The function \(g(x)=|x|+4\) is a transformation of the function \(y=|x|\), where the graph is shifted upwards by 4 units. To graph \(g(x)=|x|+4\), we note that for \(x\geq0\), \(g(x)=x+4\), and for \(x<0\), \(g(x)=-x+4\). We can choose some sample points such as, for \(x \geq 0\), \((0,4); (1, 5); (2, 6)\) etc., and for \(x < 0\), \((0,4); (-1, 5); (-2, 6)\) etc., and plot these on the graph. The graph will be V-shaped starting from the y-coordinate 4.

Step 3: Final comparison

Having both \(f(x)\) and \(g(x)\) on the same graph allows comparison. Both graphs are plotted for positive x, but \(g(x)\) also has values for negative x. Furthermore, \(g(x)\) starts at y=4, while \(f(x)\) starts at the origin. Their shapes are also different, with \(g(x)\) forming a V-shape whereas \(f(x)\) forms half a parabola. The transformation shifts the V-shaped absolute function upwards by 4 units and the V shape allows for negative x-values, unlike the square root function.

Key Concepts

Square Root Function

The square root function is a fundamental concept in graph transformations. It is represented by the function \( f(x) = \sqrt{x} \). This function is significant as it only takes non-negative values because square roots for negative numbers are not defined in the set of real numbers. The graph of \( f(x) = \sqrt{x} \) starts at the origin, which is the point where both \( x \) and \( y \) are zero. This is because \( \sqrt{0} = 0 \).

To better understand this, you can plot several points such as:

• \((0, 0)\) because \( \sqrt{0} = 0 \)
• \((1, 1)\) because \( \sqrt{1} = 1 \)
• \((4, 2)\) because \( \sqrt{4} = 2 \)
• \((9, 3)\) because \( \sqrt{9} = 3 \)

Plotting these points on the coordinate plane, the graph extends upwards to the right in a gentle curve. Understanding this graph is crucial for mastering transformations, as we often start with such basic functions before applying shifts, stretches, or reflections.

Absolute Value Function

The absolute value function, \( g(x) = |x| \), is another essential concept in graphing transformations. This function outputs the non-negative value of \( x \), meaning it measures the distance from zero. Mathematically, it's defined as:

• \( g(x) = x \) when \( x \geq 0 \)
• \( g(x) = -x \) when \( x < 0 \)

This results in a V-shaped graph centered at the origin. Each segment of the V extends straight out from the origin, creating sharp points where the graph changes direction.

For transformations like \( g(x) = |x| + 4 \), the graph is simply shifted upwards by 4 units. This is because the entire graph is lifted on the coordinate plane, modifying the position of the vertex/pivot of the V shape but preserving the overall V structure.

This type of transformation and graph is frequently encountered and forms the foundation for building more complex graph representations.

Coordinate Plane

The coordinate plane is your canvas for graph transformations like those involving square root and absolute value functions. It's a two-dimensional surface formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).

Each point on the coordinate plane is defined by a pair of numbers, known as coordinates, written as \((x, y)\). The x-coordinate tells you how far to move horizontally, while the y-coordinate tells you how far to move vertically.

The importance of the coordinate plane lies in its ability to represent graphs visually, which helps in understanding mathematical relationships and transformations. For instance:

• In the square root function, points like \((1,1)\), \((4,2)\), and \((9,3)\) can be plotted to form the graph.
• In the absolute value function, situations where \((0,4)\) or \((-2,6)\) help in shaping the V structure.

By practicing graph-plots on this plane, students visualize how functions behave and transform, enhancing their comprehension of mathematical concepts.