https://file.aiquickdraw.com/mathbot/file/1729508367411g3gnlfmw.jpg

To graph each transformed function, let's break down the transformations one by one for each function.

### 1. \( f(x) = -2|x + 1| - 5 \)
This is an absolute value function with the following transformations:
- Horizontal shift: \( x + 1 \) means a shift 1 unit to the left.
- Vertical shift: The \( -5 \) moves the graph 5 units down.
- Vertical stretch: The \( -2 \) multiplies the absolute value, flipping the graph upside down and stretching it vertically by a factor of 2.

**Vertex:** \( (-1, -5) \)

### 2. \( g(x) = (x - 1)^2 - 2 \)
This is a quadratic function with:
- Horizontal shift: \( x - 1 \) means a shift 1 unit to the right.
- Vertical shift: The \( -2 \) moves the graph 2 units down.

**Vertex:** \( (1, -2) \)

### 3. \( h(x) = \sqrt{-(x - 6)} \)
This is a square root function, which has been reflected and shifted:
- Reflection: The negative sign inside the square root reflects the graph across the y-axis.
- Horizontal shift: \( x - 6 \) moves the graph 6 units to the right.

**Starting point:** \( (6, 0) \)

### 4. \( m(x) = \sqrt{-x - 6} \)
This is also a square root function, with different transformations:
- Reflection: The \( -x \) reflects it across the y-axis.
- Horizontal shift: The \( -6 \) moves the graph 6 units left.

**Starting point:** \( (-6, 0) \)

### 5. \( n(x) = \frac{1}{3}(x + 3)^2 + 4 \)
This is a quadratic function that has been compressed and shifted:
- Horizontal shift: \( x + 3 \) moves the graph 3 units to the left.
- Vertical shift: The \( +4 \) moves the graph 4 units up.
- Vertical compression: The \( \frac{1}{3} \) compresses the graph vertically.

**Vertex:** \( (-3, 4) \)

### 6. \( p(x) = 4\sqrt{x + 5} - 1 \)
This is a square root function that has been stretched and shifted:
- Horizontal shift: \( x + 5 \) moves the graph 5 units to the left.
- Vertical shift: The \( -1 \) moves the graph 1 unit down.
- Vertical stretch: The 4 stretches the graph vertically.

**Starting point:** \( (-5, -1) \)

### 7. \( r(x) = \frac{3}{4}|x - 5| + 4 \)
This is another absolute value function:
- Horizontal shift: \( x - 5 \) moves the graph 5 units to the right.
- Vertical shift: The \( +4 \) moves the graph 4 units up.
- Vertical compression: The \( \frac{3}{4} \) compresses the graph vertically.

**Vertex:** \( (5, 4) \)

---

Would you like me to show the graphs or provide more details on any specific function?

Here are some related questions to explore:
1. How does a horizontal shift affect the shape of a function's graph?
2. What impact does a vertical stretch or compression have on the graph of a function?
3. How do absolute value transformations affect the slope of the graph?
4. How would you graph a reflection across the x-axis for any given function?
5. What is the domain and range for square root functions after transformations?

**Tip:** When graphing transformed functions, always start with the basic shape of the function (like \( |x| \), \( x^2 \), or \( \sqrt{x} \)) and apply each transformation step-by-step.

Graph a series of transformed functions including absolute value, quadratic, and square root functions. Learn about function transformations such as horizontal shifts, vertical shifts, reflections, and stretches. Suitable for high school math students in grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation