Problem 70 - College Algebra

Question

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\sqrt{x+1} $$

Step-by-Step Solution

Answer

The function \(g(x) = \sqrt{x+1}\) is the graph of the function \(f(x) = \sqrt{x}\) shifted 1 unit to the left.

Step 1: Graph the original function

Plot the graph of the original function \(f(x) = \sqrt{x}\), you can achieve this by picking a series of 'x' values, finding the corresponding 'y' values and plotting these coordinates. Remember that for the square root function, we must have \(x \geq 0\).

Step 2: Understand the transformation

The function \(g(x) = \sqrt{x+1}\) is a transformation of the function \(f(x) = \sqrt{x}\). Specifically, it's a shift transformation. It translates the graph of \(f(x)\) horizontally to the left by 1 unit. The +1 inside the square root shifts the graph to the left because it essentially 'forces' the function to start calculating the square root 1 unit earlier than it would otherwise.

Step 3: Graph the transformed function

To graph this transformation, take points from your original graph of \(f(x)\) and shift each point 1 unit to the left to form the graph of \(g(x) = \sqrt{x+1}\). Remember that for this new function, \(x \geq -1\), since the smallest value that x can take to remain in the real number domain is -1.

Key Concepts

Graphing Functions

Graphing functions is a staple in mathematics, allowing us to visually represent and analyze different mathematical relationships. When graphing functions, one begins by choosing a function, like the basic square root function, and then plotting points that satisfy the function's equation. This visually illustrates how the input values (x-values) map to output values (y-values).

To start, pick several x-values within the domain of the function. For the square root function, these x-values must be greater than or equal to zero because the square root of a negative number isn't defined in the set of real numbers. Calculate the corresponding y-values using the function's equation and plot these (x, y) coordinates on a graph.

Understanding the shape of the graph is crucial. For example, the basic graph of the square root function, \(f(x) = \sqrt{x}\), begins at the origin (0,0) and gracefully curves upwards to the right as x increases. This graph helps us intuitively grasp how the function behaves and sets the stage for any transformations we might apply.

Square Root Function

The square root function, \(f(x) = \sqrt{x}\), is fundamental in algebra and calculus. It's a non-linear function known for its distinctive half-parabolic shape. The function only takes non-negative x-values since the square root of a negative number doesn't yield a real number.

One standout feature of this function's graph is that it produces a continuous curve starting from the origin, \( (0,0) \), and extends infinitely to the right, rising gradually. The y-value or the output grows at a decreasing rate, meaning as x gets larger, the increase in y slows down.

Understanding square root functions is pivotal because they model real-world phenomena, such as calculating the magnitude of vectors in physics or interpreting non-linear scales like earthquake magnitudes. In transformations, this basic function serves as a template, making it easier to visualize shifts or reflections applied to it for more complex functions.

Horizontal Shift

Horizontal shifts are a form of transformation applied to functions to move their graphs left or right on a coordinate plane. When transforming functions like the square root function, identifying the shift direction is key to graphing the modified function accurately.

In the function \(g(x) = \sqrt{x+1}\), the "+1" inside the square root indicates a horizontal shift. Contrary to initial intuition, this shifts the graph to the **left** by 1 unit, not to the right. Why left? Because the transformation affects the input x-values directly: it adjusts each input to calculate the square root one unit earlier.

To graph this transformation, start with the original square root function \(f(x) = \sqrt{x}\). Then, shift each point of this graph 1 unit to the left. This transformation modifies the domain too. The new function is now defined for \(x \geq -1\), reflecting the leftward shift of the starting point on the x-axis.

Recognizing and graphing transformations like horizontal shifts is essential for students as they offer a deeper understanding of function behaviors and their applications.