Problem 16 - Contemporary Precalculus

Question

List the transformations needed to transform the graph of \(h(x)=2^{x}\) into the graph of the given function. $$g(x)=-5\left(2^{x-1}\right)+7$$

Step-by-Step Solution

Answer

Answer: The transformations needed include a horizontal shift of 1 unit to the right, a vertical stretch by a factor of 5, a reflection across the x-axis, and a vertical shift of 7 units up.

Step 1: Identify the base function

The base function here is h(x) = 2^x, an exponential function. We need to transform this function to get the desired function, g(x) = -5(2^(x-1)) + 7.

Step 2: Identify horizontal shift

The given function is g(x) = -5(2^(x-1)) + 7. We can see that the exponent has been modified from x to (x-1). This is a horizontal shift of 1 unit to the right. So we write it as: Horizontal Shift = 1 unit to the right.

Step 3: Identify vertical stretch/compression

Since the base function h(x) has been multiplied by -5, there is a vertical stretch by a factor of 5. This means that the graph of g(x) is 5 times taller than the graph of the base function h(x). So we write it as: Vertical Stretch/Compression = times 5.

Step 4: Identify reflection

Because the -5 in g(x) = -5(2^(x-1)) + 7 is a negative value, there is a reflection across the x-axis. This change causes the graph to be flipped over the x-axis. So we write it as: Reflection across x-axis.

Step 5: Identify vertical shift

In the given function, g(x) = -5(2^(x-1)) + 7, there is an added term of +7. This means that the graph has been shifted vertically up by 7 units. So we write it as: Vertical Shift = 7 units up. Now, putting all the transformations together, we can describe the process of transforming the graph of h(x) = 2^x into the graph of g(x) = -5(2^(x-1)) + 7 as follows: 1. Horizontal Shift: 1 unit to the right 2. Vertical Stretch/Compression: times 5 3. Reflection: across x-axis 4. Vertical Shift: 7 units up

Key Concepts

Horizontal Shift

A horizontal shift refers to the movement of a graph along the x-axis. In the transformation of the function \(h(x) = 2^x\) into \(g(x) = -5(2^{x-1}) + 7\), we notice the change in the exponent from \(x\) to \(x-1\). This indicates a horizontal shift to the right.

• The "-1" inside the exponent means we are translating 1 unit to the right.
• This shift is crucial as it affects where the graph starts along the x-axis.

This means each point on the graph of \(h(x) = 2^x\) moves to the right by 1 unit to create the graph of \(g(x)\). Remember:- Left shifts occur when there's an addition inside the function’s bracket, as in \(x + a\).- Right shifts occur with subtraction in the format of \(x - a\).Practice plotting these two functions can help in visualizing the horizontal shift transformation.

Vertical Stretch

A vertical stretch occurs when you multiply the function by a certain factor. In this instance, the original function, \(h(x) = 2^x\), is multiplied by -5, creating \(g(x) = -5(2^{x-1}) + 7\).

• This transformation scales the graph vertically by a factor of 5.
• Simply put, it makes the graph 5 times taller.

It’s important to note that this isn't just a stretch; since the factor is negative, it also implies a reflection. But here, focus on how the "5" makes each y-value 5 times further from the x-axis, enhancing the graph's vertical dimensions.Always distinguish:- Vertical stretches occur with factors greater than 1.- Vertical compressions happen when the factor is between 0 and 1.

Reflection

Reflection refers to flipping a graph across a specific axis. In the function \(g(x) = -5(2^{x-1}) + 7\), the negative sign in front of the 5 indicates a reflection over the x-axis.

• This transformation inverts points on the graph of \(h(x) = 2^x\) downwards.
• All y-values of the graph change their signs due to this flip.

Imagine placing a mirror on the x-axis – the graph will look as if it’s upside down but maintains the same shape. Reflections are pivotal when altering the general direction of the graph:- A reflection across the x-axis occurs when the entire function is multiplied by a negative.- A reflection across the y-axis happens if the input (\(x\)) is multiplied by a negative.

Vertical Shift

Vertical shifts are easier to spot as they involve adding or subtracting a constant outside the function. In the transformation from \(h(x) = 2^x\) to \(g(x) = -5(2^{x-1}) + 7\), the "+7" indicates a vertical shift upwards.

• Every point on the graph of the base function moves up by 7 units.
• It increases the height of the entire graph by a consistent amount, no matter the input \(x\).

Such vertical adjustments are fundamental in graph transformations:- Adding a positive constant pushes the graph upwards.- Subtracting a constant shifts the graph downwards.Understanding and visualizing these basic transformations can immensely help in mastering the movement of functions along the y-axis.