Problem 68 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The graph of \(g(x)=\sqrt{x}+1\) is identical to the graph of \(f(x)=\sqrt{x}\), but it is shifted one unit upwards.

Step 1: Graph the Original Function

First, graph the function \(f(x)=\sqrt{x}\). This function starts at the origin, (0,0), and increases slowly at first, then more and more rapidly as x gets large.

Step 2: Identify the Transformation

The given function \(g(x)=\sqrt{x}+1\) is actually the base function, \(f(x)=\sqrt{x}\), shifted upwards by 1 unit. This can be identified by the addition of 1 to the \(f(x)\) function.

Step 3: Apply the Transformation

To graph \(g(x)=\sqrt{x}+1\), you can take each point on the original graph of \(f(x)=\sqrt{x}\), and shift it up by 1 unit. This means if you had a point (x, y) on \(f(x)\), you would now have the point (x, y + 1) on \(g(x)\). This includes the point at the origin, which is now (0,1).

Step 4: Final Graph

Complete the graph by sketching a smooth curve through the transformed points. The graph still increases slowly at first, then more and more rapidly as x gets large, but now it starts from (0,1) instead of (0,0).

Key Concepts

Square Root Function

The square root function, represented by \(f(x) = \sqrt{x}\), serves as a fundamental building block in many mathematical contexts. The graph of this function originates at the point (0,0), and follows a distinctive pattern.

Key characteristics of the square root function include:

• The function is defined for non-negative values of \(x\), meaning \(x \geq 0\) because square roots of negative numbers are not real numbers.
• It starts at the origin and increases monotonically, meaning it keeps increasing as \(x\) increases.
• The growth of \(f(x)\) is relatively slow at first. However, as \(x\) becomes larger, the rate of increase becomes more pronounced. This produces a curve that is less steep at the beginning and steeper as \(x\) grows.

These properties make the square root function a great candidate for illustrating various graph transformations such as translations or reflections.

Vertical Shifts

A vertical shift involves moving a graph up or down along the y-axis without altering its shape. In our exercise, the transformation changes the function \(f(x) = \sqrt{x}\) to \(g(x) = \sqrt{x} + 1\).

Understanding vertical shifts is pivotal for graph transformation:

• When a constant is added to a function, \(f(x) + c\), this results in a vertical shift upwards by \(c\) units. Conversely, subtracting a constant, \(f(x) - c\), shifts the graph downwards.
• In the case of \(g(x) = \sqrt{x} + 1\), the graph of the function \(f(x) = \sqrt{x}\) is moved up by 1 unit. Each point, originally at (x, y) on \(f(x)\), moves to (x, y+1) on \(g(x)\), including the origin which moves from (0,0) to (0,1).

Vertical shifts are a common and basic transformation, making them a good starting point for students to explore graph manipulation.

Graph of Functions

Graphs are a visual representation of functions, providing insights into how the function behaves over a range of values. Constructing a graph involves plotting points that represent the equation of a function.

To plot \(f(x) = \sqrt{x}\) and \(g(x) = \sqrt{x} + 1\):

• Start by identifying points on the original graph \(f(x) = \sqrt{x}\), such as (0,0), (1,1), (4,2), etc. These points illustrate the smooth curving nature of square root functions.
• Apply the vertical shift to these points for \(g(x) = \sqrt{x} + 1\), resulting in new points: (0,1), (1,2), (4,3), etc.
• Once points are identified and plotted, connect them with a smooth curve to showcase how the function grows. This visual helps in understanding the transformation visually, which might be more intuitive than numerical manipulation alone.

Graphs are a powerful tool for students to comprehend function dynamics and exponentiate their grasp on function transformation.