Question

y= 2√x+4-3
a) state the basic function
b) state all the transformations
c) choose 3 rep. points from the basic function show each step of transformation
d) graphthe 3 points from part c)

Answer 1

To analyze the function given, \( y = 2\sqrt{x + 4} - 3 \), let's go through the steps systematically.

### a) State the basic function
The basic function is the square root function:
\[
f(x) = \sqrt{x}
\]

### b) State all the transformations
The function \( y = 2\sqrt{x + 4} - 3 \) involves the following transformations:

1. **Horizontal Shift**: The term \( \sqrt{x + 4} \) indicates a horizontal shift of \( 4 \) units to the left.
2. **Vertical Stretch**: The factor of \( 2 \) in front of the square root indicates a vertical stretch by a factor of \( 2 \).
3. **Vertical Shift**: The term \( -3 \) indicates a vertical shift downwards by \( 3 \) units.

### c) Choose 3 representative points from the basic function and show each step of transformations
Let's choose the following points from the basic function \( f(x) = \sqrt{x} \):

1. \( (0, \sqrt{0}) = (0, 0) \)
2. \( (1, \sqrt{1}) = (1, 1) \)
3. \( (4, \sqrt{4}) = (4, 2) \)

Now we will apply the transformations to each point.

1. **For the point \( (0, 0) \)**:
- Horizontal shift left 4: \( (0 - 4, 0) = (-4, 0) \)
- Vertical stretch by 2: \( (-4, 0 \cdot 2) = (-4, 0) \)
- Vertical shift down 3: \( (-4, 0 - 3) = (-4, -3) \)

2. **For the point \( (1, 1) \)**:
- Horizontal shift left 4: \( (1 - 4, 1) = (-3, 1) \)
- Vertical stretch by 2: \( (-3, 1 \cdot 2) = (-3, 2) \)
- Vertical shift down 3: \( (-3, 2 - 3) = (-3, -1) \)

3. **For the point \( (4, 2) \)**:
- Horizontal shift left 4: \( (4 - 4, 2) = (0, 2) \)
- Vertical stretch by 2: \( (0, 2 \cdot 2) = (0, 4) \)
- Vertical shift down 3: \( (0, 4 - 3) = (0, 1) \)

### Summary of transformed points
The resulting points after applying the transformations are:
1. Point A: \( (-4, -3) \)
2. Point B: \( (-3, -1) \)
3. Point C: \( (0, 1) \)

### d) Graph the 3 points from part c)
To graph the points, you would plot them on a coordinate system:

- Point A: \( (-4, -3) \)
- Point B: \( (-3, -1) \)
- Point C: \( (0, 1) \)

Here's a rough sketch of how to plot the points:

```
y
|
2| C(0, 1)
1|
0|---------------------> x
-1| B(-3, -1)
-2|
-3| A(-4, -3)
|
```

Each of these points shows the effect of the transformations on the standard square root function. The original function starts from the origin, while the transformed function is shifted left and down from the basic points chosen.