Problem 59 - College Algebra with Modeling and Visualization

Question

Use transformations to sketch a graph of \(f\). \(f(x)=1-\sqrt{x}\)

Step-by-Step Solution

Answer

Graph starts at (0,1) and decreases following the pattern of an upside-down square root.

Step 1: Identify the Base Function

The base function here is \( g(x) = \sqrt{x} \). This is the basic square root function, which starts at the origin (0,0) and increases slowly in the positive x-direction, forming half of a sideways parabola.

Step 2: Apply Vertical Reflection

The function \( g(x) \) is reflected over the x-axis. This means multiplying the whole function by -1, resulting in \( -\sqrt{x} \). This reflection flips the graph of \( \sqrt{x} \) upside-down.

Step 3: Apply Vertical Shift

Now take the function \( -\sqrt{x} \) and apply a vertical shift. We add 1 to the entire function, giving us \( 1 - \sqrt{x} \). This shifts the entire graph upward by 1 unit, moving the starting point from (0,0) to (0,1).

Step 4: Sketch the Transformed Graph

Using these transformations, we can sketch \( f(x) = 1 - \sqrt{x} \). Start at point (0,1) and follow the pattern of \( -\sqrt{x} \), where the graph descends as it moves right.

Key Concepts

Square Root Function

Understanding the square root function is essential for comprehending transformations like vertical reflections and shifts. The basic square root function, denoted as \( g(x) = \sqrt{x} \), originates at the point (0,0) on the coordinate plane. This function is a gentle curve that grows towards the right, forming what looks like half of a parabola lying on its side. The square root function only deals with non-negative values because you can only take the square root of a non-negative number in the real number system. This results in a graph that resides exclusively in the first quadrant, reflecting the fact that both \( x \) and \( \sqrt{x} \) must be non-negative. Understanding this foundation helps when applying transformations that alter its shape and position.

Vertical Reflection

Vertical reflection is a transformation that flips a graph over the x-axis. In the context of the square root function \( \sqrt{x} \), this involves multiplying the entire function by \(-1\). So, \( \sqrt{x} \) becomes \(-\sqrt{x} \). This simple multiplication affects every point on the graph, turning positive values of \( \sqrt{x} \) into their negative counterparts.

• The effect is that the graph, originally curving upwards and to the right, is now flipped over the x-axis.
• The new graph \( -\sqrt{x} \) starts at (0,0) like \( \sqrt{x} \) but descends as it moves to the right.

This reflection is crucial in understanding how our original function \( f(x) = 1 - \sqrt{x} \) forms its overall shape, as it dictates the downward direction of the curve.

Vertical Shift

A vertical shift involves moving a graph up or down on the coordinate plane. This transformation affects the function by adding (or subtracting) a constant to the entire function.

• In our example, after reflecting the square root function, the operation \( 1 - \sqrt{x} \) involves adding 1 to the reflected function \( -\sqrt{x} \).
• This addition shifts every point on the graph upward by 1 unit.
• Consequently, what used to be point (0,0) for \( -\sqrt{x} \) is now moved to (0,1).

The vertical shift is one of the simplest yet impactful transformations because it changes the starting position of the graph without affecting its shape. Combining this shift with the reflection gives us the function \( f(x) = 1 - \sqrt{x} \), which we sketch starting from the new position, curving downwards and to the right.