Problem 16 List the transformations needed ... [FREE SOLUTION]

Chapter 5: Problem 16

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

Short Answer

Expert verified

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 by step solution

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.

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.

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.

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.

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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鈥檚 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 鈥� the graph will look as if it鈥檚 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.

91影视

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

Short Answer

Step by step solution

Identify the base function

Identify horizontal shift

Identify vertical stretch/compression

Identify reflection

Identify vertical shift

Key Concepts

Horizontal Shift

Vertical Stretch

Reflection

Vertical Shift

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