Part 1 of 2

Question content area top left

Part 1

Use transformations of

f left parenthesis x right parenthesis equals x squaredf(x)=x2

to graph the following function.

h left parenthesis x right parenthesis equals left parenthesis x plus 3 right parenthesis squared plus 6h(x)=(x+3)2+6

Question content area bottom left

Part 1

Select all the transformations that are needed to graph the given function using

f left parenthesis x right parenthesis equals x squaredf(x)=x2.

A.

Shift the graph

66

unitsunits

upup.

B.

Reflect the graph about the​ x-axis.

C.

Shift the graph

66

unitsunits

downdown.

D.

Shrink the graph horizontally by a factor of

66.

E.

Reflect the graph about the​ y-axis.

F.

Stretch the graph vertically by a factor of

33.

G.

Stretch the graph horizontally by a factor of

66.

H.

Shrink the graph vertically by a factor of

33.

I.

Shift the graph

33

unitsunits

to the

leftleft.

J.

Shift the graph

33

unitsunits

to the

rightright.

...

Select all the transformations that are needed to graph the given function using f left parenthesis x right parenthesis equals x squared.

Help me solve thisView an example

Get more help

Clear all

Check answer

To graph the function \( h(x) = (x + 3)^2 + 6 \) using transformations of the basic function \( f(x) = x^2 \), let's analyze the transformations:

### Step-by-Step Transformation:
1. **Start with \( f(x) = x^2 \)**: This is the basic parabola that opens upwards with its vertex at the origin \((0, 0)\).
   
2. **Horizontal Shift**: 
   - The term \((x + 3)\) inside the squared function represents a **horizontal shift**.
   - Specifically, \(x + 3\) means that the graph is shifted **3 units to the left**.

3. **Vertical Shift**:
   - The term \(+ 6\) outside the squared function represents a **vertical shift**.
   - Specifically, adding 6 shifts the graph **6 units up**.

### Conclusion:
Based on the analysis above, the transformations needed are:
- **Shift 3 units to the left** (Option I).
- **Shift 6 units up** (Option A).

So, the correct selections are:

- **A. Shift the graph 6 units up**.
- **I. Shift the graph 3 units to the left**.

### Incorrect Options:
- **B and E (Reflect about the x-axis or y-axis)**: There is no reflection in this function.
- **C (Shift 6 units down)**: The graph is shifted up, not down.
- **D and G (Shrink/stretch horizontally)**: There is no horizontal stretching or shrinking involved.
- **F and H (Stretch/shrink vertically)**: There is no vertical stretching or shrinking.

Would you like details on the individual steps or have any further questions?

---

Here are 5 related questions that expand on this problem:

1. How would the graph change if the function was \( h(x) = -(x + 3)^2 + 6 \)?
2. What happens if we add a vertical stretch by a factor of 2 to \( h(x) = (x + 3)^2 + 6 \)?
3. How would you graph \( h(x) = (x - 3)^2 - 4 \) using transformations?
4. What is the effect of a horizontal stretch by a factor of 2 on the graph of \( f(x) = x^2 \)?
5. How would you find the vertex of the transformed function \( h(x) = (x + 3)^2 + 6 \) without graphing?

**Tip:** Remember that transformations inside the function \( f(x + c) \) affect the graph horizontally, while transformations outside \( f(x) + c \) affect the graph vertically.

Learn how to graph the quadratic function h(x) = (x + 3)^2 + 6 using transformations of the basic function f(x) = x^2. Understand horizontal and vertical shifts to accurately plot the function. Suitable for high school students studying transformations of functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation