Problem 72 - College Algebra

Question

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ k(x)=-3 \sqrt{x}-1 $$

Step-by-Step Solution

Answer

The function is a vertically stretched, reflected, and vertically shifted square root function.

Step 1: Identify the Toolkit Function

The given function is based on the square root function, which is one of the basic toolkit functions. The toolkit function is \( f(x) = \sqrt{x} \).

Step 2: Vertical Stretch and Reflection

In the equation \( k(x)=-3 \sqrt{x} \), there are two transformations applied. The factor of \(-3\) causes a vertical stretch by a factor of 3 and reflects the graph across the x-axis.

Step 3: Vertical Shift

The \(-1\) at the end of the function \( k(x)=-3 \sqrt{x}-1 \) denotes a vertical shift. This shifts the entire graph down by 1 unit.

Step 4: Sketch the Transformed Graph

Begin with the graph of \( f(x) = \sqrt{x} \), stretch it vertically by a factor of 3, reflect it across the x-axis, and then shift the entire graph down by 1 unit to sketch \( k(x) = -3 \sqrt{x} - 1 \).

Key Concepts

Vertical Stretch

In the realm of function transformations, a vertical stretch changes how a graph appears on the y-axis.
Think of pulling the graph upwards or downwards without moving it sideways. In mathematical terms, when you multiply a function by a factor greater than 1, this action stretches the function vertically.

• For example, in the function \( k(x) = -3 \sqrt{x} \), the coefficient \(-3\) is responsible for the transformation.
• The absolute value, \(3\), indicates a vertical stretch by a factor of 3.

This means every point on the graph of the original function \( \sqrt{x} \) moves three times the distance away from the x-axis.
This makes the graph look narrower as it stretches vertically.

Vertical Shift

A vertical shift, on a basic level, slides the entire graph of a function up or down.
This is dictated by a constant added or subtracted from the function. In the equation \( k(x) = -3 \sqrt{x} - 1 \), the \(-1\) at the end specifies such a movement.

• When you add a positive number, the graph shifts up.
• Conversely, subtracting a number, like \(-1\), shifts the graph down.

In our case:- Every point on the graph of the original \( \sqrt{x} \) moves down by 1 unit.
This transformation repositions the graph vertically, affecting its y-coordinates, while the shape of the graph remains unchanged.

Reflection Over X-Axis

Reflection over the x-axis is akin to flipping a pancake.
All points of the graph get mirrored across the x-axis, resulting in what was above the axis now being below, and vice versa.
In the function \( k(x) = -3 \sqrt{x} \):

• The factor \(-3\) indicates both a reflection and a vertical stretch.
• A negative sign in front of the function creates the reflection.

This means that while the graph is stretched due to the factor of 3, it also gets flipped over the x-axis.
Therefore, the shape of the function \( \sqrt{x} \) is upside-down.

Square Root Function

The square root function, \( f(x) = \sqrt{x} \), is a fundamental mathematical function.
It forms the basis for our transformation exercises. This function only takes non-negative values of \( x \) and results in non-negative outputs.
Graphically, it appears as a gentle curve starting at the origin (0,0) and moves rightwards, gradually increasing.

• Every transformation, such as stretches, shifts, or reflections, builds upon this basic shape.
• The original toolkit function helps visualize how transformations change its graph.

Transformed graphs can help solve real-world problems where certain variables depend on the square root of others.
For example, understanding how transformations like \( k(x) = -3 \sqrt{x} - 1 \) alter the original function is vital in broadening mathematical understanding and applications.