Problem 11 - 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) \(y=2 f(x+1)-3\) (b) \(y=2 f(x-1)+3\)

Step-by-Step Solution

Answer

(a) Left 1, stretch by 2, down 3. (b) Right 1, stretch by 2, up 3.

Step 1: Identify Transformations in (a)

The given function is \(y = 2f(x+1) - 3\). This function can be broken down into transformations applied successively to the graph of \(f(x)\).

Step 2: Horizontal Shift for (a)

The term \(x+1\) indicates a horizontal shift to the left by 1 unit. This is because replacing \(x\) with \(x+1\) shifts the graph left.

Step 3: Vertical Stretch for (a)

The coefficient 2 in \(2f(x+1)\) represents a vertical stretch by a factor of 2. Each \(y\)-value of \(f(x)\) is multiplied by 2, causing the graph to stretch away from the \(x\)-axis.

Step 4: Vertical Shift for (a)

Finally, the \(-3\) at the end of the expression \(2f(x+1)-3\) means the graph is shifted downward by 3 units. This decreases each \(y\)-value by 3.

Step 5: Identify Transformations in (b)

The function is \(y = 2f(x-1) + 3\), which also introduces transformations applied in a sequence from the graph of \(f(x)\).

Step 6: Horizontal Shift for (b)

The term \(x-1\) means a horizontal shift to the right by 1 unit. The \(x\)-values are increased by 1 to achieve this shift.

Step 7: Vertical Stretch for (b)

As in part (a), the multiplier 2 causes a vertical stretch by a factor of 2, extending the graph's \(y\)-values farther from the \(x\)-axis.

Step 8: Vertical Shift for (b)

The \(+3\) at the end, \(2f(x-1)+3\), indicates a shift upwards by 3 units. Each \(y\)-value is increased by 3.

Key Concepts

Horizontal Shift

When we talk about horizontal shifts in graph transformations, we're referring to moving the graph of a function left or right along the x-axis.
In the context of the problem, the function transformations described involve two types of horizontal shifts based on two expressions: \(x+1\) and \(x-1\).

• Left Shift: When you see \(x+1\) in a function, it indicates a shift of the graph to the left by 1 unit. Essentially, all x-values are decreased by 1, resulting in the graph moving left.


• Right Shift: Conversely, \(x-1\) suggests a shift to the right by 1 unit. In this case, each x-value is increased by 1, pushing the graph to the right.

Understanding these shifts is crucial, as it allows us to predict how the graph's position changes without altering its shape. The shifts are purely positional changes, meaning the graph looks the same, just at a new location on the x-axis.

Vertical Stretch

A vertical stretch transformation occurs when the y-values of a function are multiplied by a factor greater than 1. In our exercises, the multiplier is 2, which applies to both functions \(2f(x+1)\) and \(2f(x-1)\).

• The multiplier affects the y-values directly. By multiplying each y-value by 2, the graph stretches vertically, making it appear taller and steeper.


• This transformation does not change the x-values at all, so the graph's horizontal component remains unaffected. The shape becomes more elongated along the y-axis.

Vertical stretches are particularly important when analyzing how inputs are mapped to outputs within a graph.
They can dramatically change the appearance of the graph, emphasizing the vertical movements and making fluctuations more pronounced.

Vertical Shift

Vertical shifts transform the graph of a function by moving it up or down along the y-axis. This type of transformation only modifies the y-values of a function without affecting the x-values or the shape of the graph.

• Downward Shift: In the function \(2f(x+1) - 3\), the \(-3\) at the end signifies a vertical shift downwards by 3 units. This means every y-value on the graph is decreased by 3, resulting in the entire graph dropping down on the y-axis.


• Upward Shift: For the function \(2f(x-1) + 3\), the \(+3\) indicates a vertical shift upwards by 3 units. Every y-value is increased by 3, lifting the graph higher up on the y-axis.

Vertical shifts are quite straightforward, as they only require adding or subtracting a constant to the y-values.
Understanding vertical shifts helps in visualizing how a graph moves along the y-axis, whether it moves up or down in response to the function's constant addition/subtraction.