Problem 90 - College Algebra Essentials

Question

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to $$g(x)=-|x+4|+2$$

Step-by-Step Solution

Answer

The graph of function \(g(x) = -|x+4|+2\) could be obtained by reflecting the graph of \(f(x) = \sqrt{x}\) on the x-axis, shifting the graph 4 units left with respect to the x-axis, and then shifting it 2 units upwards with respect to the y-axis.

Step 1: Plot the Graph of \(f(x) = \sqrt{x}\)

Begin by sketching a simple graph of the function \(f(x) = \sqrt{x}\), the basic square root function. The square root function starts at (0,0) and curves upwards to the right.

Step 2: Identify the Transformations

Next identify the transformations that will be applied to transform \(f(x) = \sqrt{x}\) into \(g(x) = -|x+4|+2\). The transformations included in this function are: a reflection on the x-axis due to the negative sign in front of the absolute value, a horizontal shift 4 units to the left, and a vertical shift 2 units upwards.

Step 3: Apply the Reflection

Reflect the graph of \(f(x) = \sqrt{x}\) over the x-axis by changing the y-values to their opposites. The negative sign in front of the absolute value symbol implies a reflection on x-axis.

Step 4: Apply the Horizontal Shift

Shift the graph to the left by four units. The value inside the absolute value symbols will change the location of the graph on the x-axis. The term (+4) inside the absolute value means the graph will move 4 units to the left.

Step 5: Apply the Vertical Shift

Finally, shift the graph upwards by 2 units. The (+2) outside of the absolute values means that the graph will move up by two units. The graph of the function \(g(x) = -|x+4|+2\) should now be correctly positioned on the x-y grid.

Key Concepts

Square Root Function

The square root function, represented as \(f(x) = \sqrt{x}\), is one of the most basic and foundational functions in mathematics. This function maps equal pairs of numbers on both sides of zero onto the positive side of the real line.
In this graph, for positive inputs \(x\), the function produces output values that form a gentle upward curve starting at the origin (0,0) and moving to the right.
It's important to note that the function is always increasing; it never slopes downward.

• The function's domain, or the set of allowable \(x\)-values, includes zero and all positive numbers.
• The range, or the set of potential \(y\)-values, also starts at zero and extends to positive numbers.

Understanding the basic shape and properties of this function is essential before applying any transformations.

Reflection over x-axis

A reflection over the x-axis involves transforming each point on the graph so that it is correspondingly flipped over the x-axis. This means changing the sign of every \(y\)-value in the function.
Visually, it appears as if the entire graph is turned upside down. This transformation affects the range of the function by reversing its direction.

• For example, if a point on the original graph is (a, b), it becomes (a, -b) after the reflection.
• This is essential for functions needing to model negative values or other specific behaviors across the x-axis.

In our example, the function \(g(x)=-|x+4|+2\) incorporates this reflection due to the negative sign prefixed to the absolute value.

Horizontal Shift

A horizontal shift changes a graph's position along the x-axis. This occurs when a constant value is added or subtracted inside the function's formula.
This alteration shifts the entire graph either to the left or right, depending on the sign that accompanies the constant.

• If the constant is positive (e.g., \(+4\) in \(g(x) = -|x+4|+2\)), the graph shifts to the left.
• If negative, it shifts to the right.

Interpreting this transformation as a distance moved leftward is helpful. For the function \(-|x+4|\), this shift repositioning is key to realigning the graph according to the problem's requirements.

Vertical Shift

A vertical shift relocates the entire graph on the y-axis. It occurs when you add or subtract a constant value to the entire function. This transformation moves the graph up or down.
It is controlled by the constants added or subtracted outside the function's core structure.

• If a positive number is added, as in the \(+2\) in \(-|x+4|+2\), the entire graph shifts upward.
• Conversely, subtracting a number results in a downward shift.

This type of transformation doesn't affect the graph's shape or orientation but rather adjusts its position along the y-axis. In completing the graph of \(g(x) = -|x+4|+2\), this vertical shift is the final adjustment, raising the graph by two units.