Asked by unf0rgettxble

Compare the function
a (x) = x - 1.5 to the parent function f (x) = x

Vertical translation down 1.5 units
Or
Vertical translation up 1.5 units

Answers

Answered by unf0rgettxble
Compare the function c(x) = x + 100 to the same parent function as last question
Answered by unf0rgettxble
Which equation below is a quadratic function translated Left 4 units
1. Y=(x-4)^2
2. Y=x^2+4
3. Y=4x^2
4. Y=(x+4)^2
Answered by unf0rgettxble
Describe the transformation of
Y=(4+x)^2
Answered by unf0rgettxble
You're wrong it was right not left
Answered by unf0rgettxble
Compare this function to the parent function f(x)
Y=f(x+1)
Answered by unf0rgettxble
Describe the transformation from the parent quadratic function
Y=(x-2)^2
Answered by unf0rgettxble
What change does a translation cause
Answered by unf0rgettxble
Which equation below is a quadratic function translated up five units

Y=x^2+5
Y=(x+5)^2
Y=-x^2-5
Y=5x^2
Answered by unf0rgettxble
Describe the transformation of
Y=x^2+4
Answered by unf0rgettxble
What transformation transforms the graph of f(x)=x^2 to the graph of g(x)=(x+4)^2
Answered by unf0rgettxble
What's the transformation of the graph y=(x-3)^3-2
Answered by GPT-4o mini
To compare the function \( a(x) = x - 1.5 \) to the parent function \( f(x) = x \), we need to analyze how the equation for \( a(x) \) modifies the behavior of the parent function \( f(x) \).

The parent function is \( f(x) = x \), which is a straight line passing through the origin with a slope of 1.

In the function \( a(x) = x - 1.5 \), the term \(-1.5\) indicates a vertical transformation. Specifically, since we are subtracting 1.5 from \( x \), this results in a vertical translation of the graph downward by 1.5 units.

Therefore, the correct interpretation is:

**Vertical translation down 1.5 units.**

The graph of \( a(x) \) will look the same as the graph of \( f(x) \), but it will be shifted downwards by 1.5 units.
Answered by GPT-4o mini
To compare the function \( c(x) = x + 100 \) to the parent function \( f(x) = x \), we can analyze how the expression for \( c(x) \) modifies the parent function.

The parent function is again \( f(x) = x \).

In the function \( c(x) = x + 100 \), the term \( +100 \) indicates a vertical transformation. Specifically, because we are adding 100 to \( x \), this results in a vertical translation of the graph upward by 100 units.

Therefore, the correct interpretation is:

**Vertical translation up 100 units.**

The graph of \( c(x) \) will look the same as the graph of \( f(x) \), but it will be shifted upwards by 100 units.
Answered by GPT-4o mini
To determine which equation represents a quadratic function translated left by 4 units, we need to understand how translations work. A translation to the left by \( h \) units in a quadratic function \( f(x) = (x - k)^2 \) is represented by \( f(x) = (x + h)^2 \).

For a left translation of 4 units, we would have:

- The standard form for a translation left would be \( (x + 4)^2 \).

Now, let's evaluate the options:

1. \( Y = (x - 4)^2 \) - This represents a translation **right** by 4 units.
2. \( Y = x^2 + 4 \) - This is a vertical translation up by 4 units and does not represent a leftward shift.
3. \( Y = 4x^2 \) - This is a vertical scaling, not a translation.
4. \( Y = (x + 4)^2 \) - This represents a translation **left** by 4 units.

Thus, the equation that represents a quadratic function translated left by 4 units is:

**4. \( Y = (x + 4)^2 \)**.
Answered by GPT-4o mini
The function \( Y = (4 + x)^2 \) can be transformed to highlight its relationship to the parent function \( f(x) = x^2 \).

1. **Rewrite the function**: First, we can rewrite \( Y = (4 + x)^2 \) as \( Y = (x + 4)^2 \). This helps us see the transformation more clearly.

2. **Identify the transformation**: In the form \( (x + h)^2 \), where \( h \) is a positive number, this indicates a horizontal translation. Specifically, the expression \( (x + 4) \) indicates a translation **to the left** by 4 units.

3. **Vertical stretch/compression**: There is no additional coefficient outside of the squared term, so there are no vertical stretches or compressions applied here.

4. **Vertical translations**: There is also no constant added or subtracted outside the squared term, meaning there is no vertical translation.

### Summary of the Transformation:
- **Horizontal Translation**: Left 4 units.

Thus, the transformation of the function \( Y = (4 + x)^2 \) is a horizontal translation **left by 4 units** from the parent function \( f(x) = x^2 \).
Answered by GPT-4o mini
I apologize for any confusion. Let's clarify the transformation of the function \( Y = (4 + x)^2 \).

When we rewrite the expression as \( Y = (x + 4)^2 \), we can analyze it in the context of the parent function \( f(x) = x^2 \):

1. **Formulation**: The expression \( (x + 4) \) indicates that for every \( x \), we are substituting \( x \) with \( x + 4 \).

2. **Translation Direction**:
- A positive addition within the parentheses (as in \( (x + 4) \)) means to shift the graph **to the left**.
- Conversely, if we had \( (x - 4) \), it would shift the graph **to the right**.

### Conclusion:
- The function \( Y = (4 + x)^2 \) indeed reflects a horizontal translation **to the left by 4 units**.

If there was a misunderstanding about the terms "right" vs. "left," I’m here to help clarify further! Please let me know how I can assist you.
Answered by GPT-4o mini
To compare the function \( Y = f(x + 1) \) to the parent function \( f(x) \), we can analyze how this transformation affects the original function.

### Parent Function
The parent function \( f(x) \) is simply the basic form of a function (for example, it could be \( f(x) = x^2 \), \( f(x) = x \), etc., depending on the context).

### Transformation Explanation
1. **Function Form**: The expression \( f(x + 1) \) indicates that we are replacing \( x \) in the function \( f(x) \) with \( (x + 1) \).

2. **Effect of the Transformation**:
- When we have a transformation of the form \( f(x + h) \), where \( h \) is a positive number, it results in a **horizontal translation to the left** by the value of \( h \).
- In this case, since \( h = 1 \), the function is translated **left by 1 unit**.

### Summary:
- The function \( Y = f(x + 1) \) represents a **horizontal translation of the graph of \( f(x) \) left by 1 unit**.

This means that whichever points are on the graph of the original parent function \( f(x) \), they will now appear 1 unit to the left on the graph of \( Y = f(x + 1) \).
Answered by GPT-4o mini
To describe the transformation of the function \( Y = (x - 2)^2 \) from the parent quadratic function \( f(x) = x^2 \), let's break down the changes:

### Parent Function
- The parent quadratic function is \( f(x) = x^2 \). This is a standard parabola that opens upwards and has its vertex at the origin (0, 0).

### Transformation Analysis
1. **Rewrite the Function**: The function can be expressed as \( Y = (x - 2)^2 \).

2. **Identify the Transformation**:
- The expression within the parentheses, \( (x - 2) \), indicates a horizontal translation.
- Specifically, a subtraction inside the parentheses means that we shift to the **right** by 2 units.

3. **No Vertical Changes**: There are no additional coefficients or constants outside the squared term, so:
- There is no vertical stretch or compression.
- There is no vertical translation (upward or downward).

### Summary of Transformation
- The function \( Y = (x - 2)^2 \) represents a **horizontal translation to the right by 2 units** from the parent quadratic function \( f(x) = x^2 \).

### Conclusion
Thus, the transformation from the parent function \( f(x) = x^2 \) to \( Y = (x - 2)^2 \) moves the graph of the quadratic to the right by 2 units. The vertex of the parabola will move from (0, 0) to (2, 0).
Answered by GPT-4o mini
A translation in the context of functions or graphs refers to shifting the entire graph of a function in a specific direction without altering its shape or orientation. Here are the key aspects of how translations cause changes to functions:

### Types of Translations

1. **Horizontal Translation**:
- **Definition**: This involves shifting the graph left or right.
- **Horizontal Shift**:
- If the function is of the form \( f(x - h) \), where \( h > 0 \), the graph shifts **to the right** by \( h \) units.
- If the function is of the form \( f(x + h) \), it shifts **to the left** by \( h \) units.
- **Example**: For \( Y = f(x - 2) \), the graph shifts 2 units to the right. For \( Y = f(x + 3) \), it shifts 3 units to the left.

2. **Vertical Translation**:
- **Definition**: This refers to shifting the graph up or down.
- **Vertical Shift**:
- If the function is of the form \( f(x) + k \), where \( k > 0 \), the graph shifts **upward** by \( k \) units.
- If the function is of the form \( f(x) - k \), it shifts **downward** by \( k \) units.
- **Example**: For \( Y = f(x) + 5 \), the graph shifts up 5 units. For \( Y = f(x) - 3 \), it shifts down 3 units.

### Properties of Translations
- **Shape**: The shape and orientation of the graph remain unchanged during a translation. For example, if \( f(x) \) is a parabola, translating it horizontally or vertically will still yield a parabola of the same shape.
- **Points**: All points on the graph are moved. For example, if a point \( (x, y) \) on the original graph moves due to a translation to \( (x + h, y + k) \), all points shift in the same manner.
- **Function Values**: The function values (outputs) change according to the translation but remain consistent in their relation to the inputs (horizontal position).

### Summary
In conclusion, a translation affects the position of the graph of a function by shifting it left, right, up, or down without altering its shape or size. This makes translations a fundamental type of transformation in understanding the behavior of functions.
Answered by GPT-4o mini
To identify which equation represents a quadratic function translated up 5 units, we need to look for an equation that can be expressed in the form of the parent quadratic function \( f(x) = x^2 \) with an additional constant that shifts the graph upward.

### Analyze the Options:

1. **\( Y = x^2 + 5 \)**: This is the equation of a quadratic function with a vertical translation upward by 5 units because we are adding 5 to the output of the parent function. **This is correct**.

2. **\( Y = (x + 5)^2 \)**: This represents a horizontal translation to the left by 5 units. This does not translate the graph upward. **Incorrect**.

3. **\( Y = -x^2 - 5 \)**: This equation represents a downward-opening parabola with a vertical translation downward by 5 units. **Incorrect**.

4. **\( Y = 5x^2 \)**: This is a vertically stretched version of the parent function \( f(x) = x^2 \) but does not involve a vertical translation. **Incorrect**.

### Conclusion:
The equation that represents a quadratic function translated **up** 5 units is:

**\( Y = x^2 + 5 \)**.
Answered by GPT-4o mini
To describe the transformation of the function \( Y = x^2 + 4 \) compared to its parent quadratic function \( f(x) = x^2 \), we can analyze how the equation modifies the behavior of the graph.

### Parent Function
- The parent function \( f(x) = x^2 \) is a standard upward-opening parabola with its vertex at the origin (0, 0).

### Transformation Analysis
1. **Vertical Shift**:
- In the equation \( Y = x^2 + 4 \), we are adding 4 to the output of the parent function \( f(x) \).
- This means that the entire graph of \( f(x) \) is shifted **upward** by 4 units.

2. **No Horizontal Changes**:
- There are no changes made inside the parentheses; thus, there is no horizontal translation (left or right). The expression is still \( x^2 \).

3. **No Vertical Stretch/Compression**:
- There is no coefficient multiplying the \( x^2 \) term (like 2 or \(\frac{1}{2}\)), so there is no vertical stretch or compression.

### Summary of the Transformation
- The graph of \( Y = x^2 + 4 \) represents a **vertical translation upward by 4 units** from the parent function \( f(x) = x^2 \).

### Conclusion
Thus, the transformation from the parent function \( f(x) = x^2 \) to \( Y = x^2 + 4 \) is a **vertical shift up by 4 units**. The vertex of the parabola shifts from (0, 0) to (0, 4).
Answered by GPT-4o mini
To describe the transformation that changes the graph of the parent quadratic function \( f(x) = x^2 \) to the graph of \( g(x) = (x + 4)^2 \), let's analyze the change in the equation.

### Analysis of the Functions

1. **Parent Function**:
- The parent function is \( f(x) = x^2 \), which is a parabola that opens upwards with its vertex at the origin (0, 0).

2. **Transformed Function**:
- The transformed function is \( g(x) = (x + 4)^2 \).

### Transformation Description

- The expression \( (x + 4) \) indicates a horizontal translation.
- Specifically, in the form \( f(x + h) \):
- If the equation is \( f(x - h) \) where \( h \) is positive, the graph shifts **to the right**.
- If it is \( f(x + h) \) where \( h \) is positive, the graph shifts **to the left**.

In this case, since we have \( (x + 4) \), it indicates a horizontal translation **to the left by 4 units**.

### Summary
To summarize, the transformation that changes the graph of \( f(x) = x^2 \) to \( g(x) = (x + 4)^2 \) is a **horizontal translation to the left by 4 units**. The vertex of the parabola shifts from (0, 0) to (-4, 0).
Answered by GPT-4o mini
To describe the transformation of the function \( y = (x - 3)^3 - 2 \) compared to its parent cubic function \( f(x) = x^3 \), we can analyze the changes made to the equation.

### Parent Function
- The parent cubic function is \( f(x) = x^3 \), which has a characteristic shape and passes through the origin (0, 0).

### Analyzing the Transformed Function
1. **Horizontal Translation**:
- The term \( (x - 3) \) indicates that the graph of the parent function \( f(x) = x^3 \) is shifted horizontally.
- Specifically, since we are subtracting 3 from \( x \), this results in a horizontal translation **to the right by 3 units**.

2. **Vertical Translation**:
- The term \( -2 \) outside the cubic function indicates that the entire graph is shifted vertically.
- Since we are subtracting 2, the graph will shift **downward by 2 units**.

### Summary of Transformations
Combining both transformations, we have:
- **Horizontal Translation**: Right by 3 units.
- **Vertical Translation**: Down by 2 units.

### Conclusion
Thus, the transformation that changes the graph of the parent cubic function \( f(x) = x^3 \) to the graph of \( y = (x - 3)^3 - 2 \) includes a **horizontal shift to the right by 3 units** and a **vertical shift downward by 2 units**. The new vertex of the transformed function would be at the point \( (3, -2) \).

Related Questions