Problem 29 - Algebra and Trigonometry

Question

\(27-32\) : A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=\sqrt{x} ;\) shift 3 units to the left, stretch vertically by a factor of \(5,\) and reflect in the \(x\) -axis

Step-by-Step Solution

Answer

The transformed function is \( f(x) = -5\sqrt{x + 3} \).

Step 1: Shift 3 Units to the Left

To shift the graph of the function \( f(x) = \sqrt{x} \) 3 units to the left, we replace \( x \) with \( x + 3 \) in the function. Thus, the transformed function after this step is \( f(x) = \sqrt{x + 3} \).

Step 2: Stretch Vertically by a Factor of 5

To stretch the graph vertically by a factor of 5, we multiply the entire function by 5. So the function from Step 1 becomes \( f(x) = 5\sqrt{x + 3} \).

Step 3: Reflect in the x-axis

To reflect a function in the \( x \)-axis, we multiply the entire function by \(-1\). Therefore, the function from Step 2 becomes \( f(x) = -5\sqrt{x + 3} \).

Key Concepts

Vertical Stretch

In function transformations, a vertical stretch involves making the graph of the function taller. This is performed by multiplying the entire function by a constant factor greater than one. In the case of our function, \( f(x) = \sqrt{x} \), the vertical stretch factor is 5. This means you multiply the function by 5, making it \( 5\sqrt{x+3} \).
Such a transformation stretches all the points on the graph away from the x-axis:

• If the original function contained a point (a,b), the stretched function would move that point up to (a,5b) if b is positive.
• The stretch only affects the y-values of the function, not the x-values.

Imagine a rubber band getting pulled up while being pinned at the x-axis, that’s a vertical stretch! It maintains the graph's overall shape but enlarges it vertically.

Horizontal Shift

A horizontal shift in functions occurs when the graph moves left or right. For the function \( f(x) = \sqrt{x} \), shifting it to the left by 3 units involves replacing \( x \) with \( x+3 \), resulting in \( f(x) = \sqrt{x+3} \).
Horizontal shifts are intuitive changes:

• Shifting left means you add a positive number inside the function's argument, e.g., \( x+3 \).
• To shift right, you'd subtract from the x inside the function, like \( x-3 \).
• The entire graph slides without changing its shape.

This transformation allows the function to start its behavior under different "circumstances," perhaps exposing features not visible in its original form. It's important to notice such movements during transformations because they are about changing the input criteria rather than the resulting values.

Reflection Across the X-Axis

Reflecting a graph over the x-axis might sound complex but it's quite simple. This transformation will flip the graph upside down. For the function in question, \( f(x) = 5\sqrt{x+3} \), reflecting it over the x-axis involves multiplying the function by \(-1\), changing it to \( -5\sqrt{x+3} \).
Here's what happens during this reflection:

• All positive y-values of the function become negative, and vice versa.
• If a point (a,b) lies on the original graph, the reflected point will be (a,-b) for the transformed graph.

If you imagine the graph as a mountain, reflection across the x-axis turns it into a valley. This is how you can visually invert the function's output, changing all heights and depths to their opposites.