Problem 30 - Beginning and Intermediate Algebra

Question

Sketch the graph of \(f(x) .\) Then, graph \(g(x)\) on the same axes using the transformation techniques. $$\begin{aligned}&f(x)=\sqrt{x-2}\\\&g(x)=-\sqrt{x-2}\end{aligned}$$

Step-by-Step Solution

Answer

Sketch the graph of \(f(x) = \sqrt{x-2}\) by plotting the point (2, 0) as the starting point and creating a curve that increases as x increases with domain \(x \ge 2\) and range \(y \ge 0\). For \(g(x) = -\sqrt{x-2}\), apply a vertical reflection to the graph of \(f(x)\). This means the graph of \(g(x)\) starts at point (2, 0) and creates a curve that decreases as x increases with domain \(x \ge 2\) and range \(y \le 0\).

Step 1: Sketch the graph of \(f(x) = \sqrt{x-2}\).

To sketch the graph of \(f(x) = \sqrt{x-2}\), we first identify the important features of this square root function: 1. The domain: \(x \ge 2\), since we can only take the square root of a non-negative number. 2. The range: \(y \ge 0\), since the square root function is always non-negative. 3. The graph shape: It will be a square root function, starting from the point (2, 0) and increasing as x increases. To sketch the graph, plot the point (2, 0) as the starting point, and create a curve that increases as x increases and stays within the domain and range we just identified.

Step 2: Identify the transformation to apply to the graph of \(f\).

We want to find the graph of \(g(x) = -\sqrt{x-2}\). We can see that the difference between this function and the original function is the negative sign in front of the square root. This negative sign indicates that we will apply a vertical reflection of the graph of \(f\) to find the graph of \(g\).

Step 3: Apply the vertical reflection to the graph of \(f\).

To apply the vertical reflection, we need to change the y-coordinate of each point on the graph of \(f\) to its negative value. This means that points above the x-axis on the graph of \(f\) will be reflected to become points below the x-axis on the graph of \(g\), and vice versa. We already know that the graph of \(f\) starts at point (2, 0), which will also be the starting point of the graph of \(g\). From this point, draw the graph of g by creating a curve that decreases as x increases and stays within the domain \(x \ge 2\) and the updated range \(y \le 0\). The graph of \(g(x) = -\sqrt{x-2}\) is the vertical reflection of the graph of \(f(x) = \sqrt{x-2}\) about the x-axis.

Key Concepts

Domain and Range

In mathematics, understanding the domain and range of a function is essential to fully grasp its graph and transformations. The domain refers to the set of all possible input values (or x-values) that a function can accept without causing any issues like division by zero or taking the square root of a negative number.
The given function is a square root function, specifically, \(f(x) = \sqrt{x-2}\). To find its domain, we consider the expression under the square root: \(x-2\).

• The expression \(x-2\) must be greater than or equal to zero for real numbers.
• Thus, solving the inequality \(x-2 \geq 0\), we find \(x \geq 2\).

The range of a function is the set of possible output values (or y-values). For \(f(x) = \sqrt{x-2}\), the range consists of all non-negative real numbers because square roots yield non-negative results by nature. As such:

• For \(f(x)\), the range is \(y \geq 0\), resulting in outputs starting from \(0\) and going upwards.

These constraints define where on the graph our function will plot points, ensuring those values fall within allowable limits.

Vertical Reflection

A vertical reflection involves flipping a graph over the x-axis, essentially changing the orientation of the graph without altering its shape. In our exercise, we look at how to transform \(f(x) = \sqrt{x-2}\) into \(g(x) = -\sqrt{x-2}\) through this process.
The negative sign placed in front of the square root function signals us to perform a vertical reflection.

• This transformation changes every y-coordinate \(f(x)\) of the original graph to its negative counterpart \(-f(x)\).
• For example, a point \((a, b)\) on \(f(x)\) will become \((a, -b)\) on \(g(x)\).

This vertical reflection causes the graph of \(g(x)\) to start at (2, 0), similar to \(f(x)\), as this point lies exactly on the x-axis, hence it does not move. All other points will mirror below the x-axis, creating a downward-opening curve while adhering to the domain \(x \ge 2\) and range \(y \le 0\). Reflecting a graph can illustratively provide you double viewpoints into how functions operate at different values.

Square Root Function

Square root functions form a unique category of functions characterized by their non-linear "half parabola" shape. The standard form is \(f(x) = \sqrt{x} \), but transformations like \(f(x) = \sqrt{x-2} \) shift or modify their behavior on the graph.
Key properties include:

• Starting point or vertex. For \(f(x) = \sqrt{x-2}\), the function begins at (2, 0), dictated by setting the radicand \(x-2\) to zero.
• The graph will increase gradually, reflecting the square root function's slowing growth as x increases.

When transforming functions, symbols, and operations outside the square root can affect both the graph's position and orientation. For instance, a negative coefficient as in \(-\sqrt{x-2}\) leads to flipping the graph vertically.
This causes the familiar upward curve of a square root function to shift to a downward curve for \(g(x) = -\sqrt{x-2}\), all while originating from the same x-value start point (2, 0).
Understanding how changes to the basic square root function affect its graph is pivotal to mastering function transformations, ensuring readiness for more complex mathematical concepts.