Problem 6 - Algebra and Trigonometry

Question

1–10 ? Suppose the graph of f is given. Describe how the graph of each function can be obtained from the graph of f. (a) \(y=-f(x)+5 \quad\) (b) \(y=3 f(x)-5\)

Step-by-Step Solution

Answer

(a) Reflect over x-axis, then shift up by 5 units; (b) Stretch vertically by 3, then shift down by 5 units.

Step 1: Understanding the problem

The problem is about how transformations affect a graph. We need to describe how graphs of new functions can be derived from an initial graph of a base function, \( f(x) \).

Step 1: Analyzing the function y = -f(x) + 5

Start with \(y = -f(x) + 5\). The 'minus' sign in front of \(f(x)\) indicates a reflection over the x-axis. The '+5' represents a vertical shift upwards by 5 units.

Step 2: Applying transformations to y = -f(x) + 5

To graph \(y = -f(x) + 5\): First, take the graph of \(f(x)\) and reflect it over the x-axis, creating the graph of \(-f(x)\). Then shift this graph up by 5 units.

Step 3: Analyzing the function y = 3f(x) - 5

Now, consider \(y = 3f(x) - 5\). The '3' indicates a vertical stretch by a factor of 3. The '-5' is a vertical shift downwards by 5 units.

Step 4: Applying transformations to y = 3f(x) - 5

To graph \(y = 3f(x) - 5\): First, vertically stretch the graph of \(f(x)\) by a factor of 3, producing the graph of \(3f(x)\). Then shift this graph down by 5 units.

Key Concepts

Reflection over the x-axis

In graph transformations, reflecting a graph over the x-axis changes the position of the graph by flipping it vertically. This transformation is denoted by placing a negative sign in front of the function, like \-f(x)\. For instance, when you reflect the graph of \(f(x)\) over the x-axis, each positive value of \(f(x)\) in the original graph becomes negative, and vice versa. This results in a mirror image of the graph across the x-axis.
To reflect, follow these steps:

• Identify all points on the original graph.
• Flip each point vertically across the x-axis.
• Negative points become positive and positive points become negative.

Reflection over the x-axis often comes first in multi-transformation problems. This transformation does not affect the shape of the graph, only its orientation.

Vertical Shift

A vertical shift involves moving the entire graph of a function up or down without altering its shape. This transformation is determined by adding or subtracting a constant from the function. For example, the expression \(f(x) + k\) shifts the graph of \(f(x)\) upwards by \(k\) units, while \(f(x) - k\) shifts it downwards by \(k\) units.
Vertical shifts are simple and easy to execute:

• Moving up means adding a positive constant to every output \(f(x)\) value.
• Moving down requires subtracting a constant from each \(f(x)\) value.

After determining the direction (up or down) and the distance, just move all points of the graph uniformly along the vertical axis. This shift does not change the graph's orientation or shape.

Vertical Stretch

A vertical stretch changes the shape of the graph by stretching it away from or compressing it towards the x-axis. This type of transformation is indicated by multiplying the function by a coefficient larger or smaller than 1. For instance, in \(3f(x)\), the graph of \(f(x)\) is stretched vertically by a factor of 3.
When performing a vertical stretch:

• Multiply each \(y\)-value of \(f(x)\) by the stretch factor.
• If the factor is more than 1, the graph stretches and points move away from the x-axis.
• If the factor is between 0 and 1, the graph compresses and points move toward the x-axis.

This transformation affects only the y-values, making the graph taller or shorter. It’s critical to apply this transformation before shifting vertically when combining transformations.