Question

3.7 practice with calcchat and calcview
in exercises 1 - 6, use the graphs of f and g to describe the transformation from the graph of f to the graph of g. see example 1.

1. g(x)=f(x)+2 vertical shift up 2
2. g(x)=f(x)-3 vertical shift down by 3 units
3. g(x)=f(x)-3 vertical down shift
4. f(x)=-3x + 4;g(x)=f(x)+1 horizontal shift left by 5 units
5. f(x)=-x - 2;g(x)=f(x + 5) horizontal shift left by 5 units
6. f(x)=1/2x - 5;g(x)=f(x - 3) horizontal shift right by 3 units
7. modeling real life you and your friend start biking from the same location. your distance (in miles) after t minutes is represented by d(t)=1/2t. your friend starts biking 5 minutes after you. her distance is represented by f(t)=d(t - 5). describe the transformation from the graph of d to the graph of f horizontal shift right by 5 units friend starts 5 minutes later.
8. modeling real life the total cost (in dollars) to bowl n games is represented by c(n)=4.5n + 2.5. the shoe - rental price increases $0.50. the new total cost is represented by t(n)=c(n)+0.5. describe the transformation from the graph of c to the graph of t vertical shift up by 0.50 shoe rental price increase

Explanation:

Step1: Recall function - transformation rules

For a function \(y = f(x)\), a vertical shift is of the form \(y=f(x)+k\) (up if \(k>0\), down if \(k < 0\)) and a horizontal shift is of the form \(y = f(x - h)\) (right if \(h>0\), left if \(h < 0\)).

Step2: Analyze each case

Case 1: \(g(x)=f(x)+a\)

If \(a = 2\) (in \(g(x)=f(x)+2\)), it is a vertical shift up by 2 units. If \(a=-3\) (in \(g(x)=f(x)-3\)), it is a vertical shift down by 3 units.

Case 2: \(g(x)=f(x + b)\)

If \(b = 5\) (in \(g(x)=f(x + 5)\)), it is a horizontal shift left by 5 units. If \(b=-3\) (in \(g(x)=f(x-3)\)), it is a horizontal shift right by 3 units.

Case 3: Real - life examples

For \(f(t)=d(t - 5)\) where \(d(t)=\frac{1}{2}t\), since it is of the form \(y = f(x - h)\) with \(h = 5\), it is a horizontal shift right by 5 units. For \(T(n)=C(n)+0.5\) where \(C(n)=4.5n + 2.5\), since it is of the form \(y=f(x)+k\) with \(k = 0.5\), it is a vertical shift up by 0.5 units.

Answer:

The solutions provided in the hand - written work are correct. The transformations are as follows:

1. \(g(x)=f(x)+2\) is a vertical shift up by 2 units.
2. \(g(x)=f(x)-3\) is a vertical shift down by 3 units.
3. \(g(x)=f(x)+1\) is a vertical shift up by 1 unit.
4. \(g(x)=f(x)+1\) (assuming the correct form for transformation analysis) is a vertical shift up by 1 unit.
5. \(g(x)=f(x + 5)\) is a horizontal shift left by 5 units.
6. \(g(x)=f(x-3)\) is a horizontal shift right by 3 units.
7. \(f(t)=d(t - 5)\) is a horizontal shift right by 5 units.
8. \(T(n)=C(n)+0.5\) is a vertical shift up by 0.5 units.