Problem 9 - Precalculus Mathematics for Calculus

Question

Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f\). (a) \(y=-f(x)+5\) (b) \(y=3 f(x)-5\)

Step-by-Step Solution

Answer

Reflect, shift up for (a); stretch, shift down for (b).

Step 1: Reflect across the x-axis

To obtain the graph of \(y = -f(x)\) from \(f(x)\), reflect it across the x-axis. This means that for every point \((x, y)\) on \(f\), there will be a corresponding point \((x, -y)\) on the graph of \(-f(x)\).

Step 2: Vertical Shift Up

Shift the graph of \(-f(x)\) upwards by 5 units to obtain the graph of \(y = -f(x) + 5\). This means adding 5 to the y-coordinate of each point on \(-f(x)\), resulting in a new point \((x, -y + 5)\).

Step 3: Vertical Stretch

For \(y = 3f(x)\), multiply each y-coordinate of the graph of \(f(x)\) by 3. This stretches the graph vertically by a factor of 3. Each point \((x, y)\) on \(f(x)\) becomes \((x, 3y)\).

Step 4: Vertical Shift Down

After performing the vertical stretch, shift the graph down by 5 units to obtain \(y = 3f(x) - 5\). Subtract 5 from each y-coordinate of the stretched graph, resulting in the final points \((x, 3y - 5)\).

Key Concepts

Reflection

Reflection is a transformation that flips a graph over a specific line. When dealing with functions, a common type of reflection is across the x-axis.
To reflect a function across the x-axis, you change the sign of the y-values in the original graph. This transformation can be represented by the equation:

• If your function is \(f(x)\), the reflection across the x-axis is \(y = -f(x)\).

Imagine every point on the graph of \(f(x)\) is mirrored over the x-axis to create \(-f(x)\). For example, a point \((x, y)\) on the function will become \((x, -y)\) after reflection.
This type of transformation is useful when you want to visually reverse the direction of the graph along the vertical axis. It can change peaks into troughs and vice versa.

Vertical Shift

A vertical shift involves moving the entire graph up or down along the y-axis. This transformation does not affect the shape of the graph, only its position. Vertical shifts are achieved by adding or subtracting a constant from the function:

• To shift up by \(c\) units, the equation becomes \(y = f(x) + c\).
• To shift down by \(c\) units, use \(y = f(x) - c\).

For instance, if you have reflected your function to get \(-f(x)\), you can shift it up by 5 units through the equation \(y = -f(x) + 5\).
This moves every point \((x, -y)\) on the reflected graph up to \((x, -y + 5)\). Vertical shifts make it easy to adjust the height of a graph without altering its overall form.

Vertical Stretch

A vertical stretch transformation involves expanding or compressing the graph vertically. This changes the graph's height while maintaining its horizontal position.
A vertical stretch is represented mathematically by multiplying each y-coordinate by a factor. For example:

• A stretch by a factor of 3 is shown by \(y = 3f(x)\).

When stretching a graph, a point \((x, y)\) becomes \((x, 3y)\). This transformation makes peaks taller and valleys deeper.
However, if you wish to change the vertical position after stretching, a vertical shift can be applied. For example, by following a vertical stretch with a downward shift of 5 units, your function would look like this: \(y = 3f(x) - 5\).
Vertical stretches are powerful for amplifying certain features of a graph, making it easier to analyze patterns or features that depend on magnitude.