Problem 11 - Algebra and Trigonometry

Question

\(5-14\) . Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f .\) (a) \(y=2 f(x+1)-3 \quad\) (b) \(y=2 f(x-1)+3\)

Step-by-Step Solution

Answer

(a) Shift left 1, stretch vertically by 2, and move down 3. (b) Shift right 1, stretch vertically by 2, and move up 3.

Step 1: Identify Transformations for Part (a)

For the function given in part (a), \(y = 2f(x+1) - 3\), let's identify the transformations one by one:1. **Horizontal Shift**: The \((x+1)\) represents a horizontal shift. Here, \(+1\) means shifting the graph left by 1 unit.2. **Vertical Stretch**: The coefficient 2 in front of \(f(x)\) indicates a vertical stretch, multiplying the \(y\)-values by 2.3. **Vertical Shift**: The \(-3\) at the end indicates a vertical shift downwards by 3 units.

Step 2: Apply Transformations for Part (a)

Now apply each transformation identified:1. Start with the graph of \(f(x)\).2. Move the entire graph 1 unit to the left to account for \(x+1\).3. Stretch the graph vertically by a factor of 2.4. Finally, shift the graph downward by 3 units.

Step 3: Identify Transformations for Part (b)

For the function given in part (b), \(y = 2f(x-1) + 3\), identify the transformations:1. **Horizontal Shift**: The \((x-1)\) indicates a horizontal shift. Here, \(-1\) means shifting the graph right by 1 unit.2. **Vertical Stretch**: The coefficient 2 in front of \(f(x)\) still represents a vertical stretch, multiplying the \(y\)-values by 2.3. **Vertical Shift**: The \(+3\) indicates a vertical shift upwards by 3 units.

Step 4: Apply Transformations for Part (b)

Now apply each transformation:1. Start with the graph of \(f(x)\).2. Shift the entire graph 1 unit to the right to account for \(x-1\).3. Stretch the graph vertically by a factor of 2.4. Finally, shift the graph upwards by 3 units.

Key Concepts

Horizontal Shift

A horizontal shift involves moving the entire graph of a function left or right on the coordinate plane. This occurs when we add or subtract a constant from the input variable, typically the "x" inside the function. For example, in the function \(y = f(x + 1)\), the \(+1\) indicates a horizontal shift. But it can be a bit tricky: you move the graph _opposite_ to the sign inside the parentheses.

• If \(f(x + c)\), the graph shifts left by \(c\) units.
• If \(f(x - c)\), it shifts right by \(c\) units.
<\ul>In our example, \(y = 2f(x+1) - 3\) results in shifting the graph left by 1 unit, while \(y = 2f(x-1) + 3\) moves it right by 1 unit. Remember, dealing with horizontal shifts means being extra attentive to the signs!

Vertical Stretch

A vertical stretch transformation affects how the points of our graph expand either towards or away from the x-axis. This happens when we multiply the function by a constant greater than 1. If the constant is less than 1 but greater than 0, it instead causes a vertical compression.

• If the coefficient \(a\) in front of \(f(x)\) is greater than 1, like \(y = 2f(x)\), every \(y\)-value is multiplied by \(a\), making the graph taller and narrower.
• If \(0 < a < 1\), like \(a = 0.5\), the graph compresses vertically.

In the examples \(y = 2f(x+1) - 3\) and \(y = 2f(x-1) + 3\), the coefficient 2 stretches both graphs vertically by a factor of 2. Each point on the graph is twice as high, making the graph "zoomed out" in terms of height.

Vertical Shift

A vertical shift changes the graph by moving it up or down without altering its shape. This occurs when we add or subtract a constant from the whole function. If you add a constant, every point on the graph moves up; subtract, and each point moves down.

• The expression \(+ c\) causes an upward shift by \(c\) units.
• The expression \(- c\) causes a downward shift by \(c\) units.
<\ul>In our examples, the function \(y = 2f(x+1) - 3\) shifts down by 3 units due to the \(-3\), while \(y = 2f(x-1) + 3\) shifts up by 3 units because of the \(+3\). This shift affects the entire graph consistently, raising or lowering all points equally across the board.