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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

23 answers

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.

Compare the function c(x) = x + 100 to the same parent function as last question

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.

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

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. \( Y = (x + 4)^2 \).

Describe the transformation of
Y=(4+x)^2

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 \( f(x) = x^2 \).

You're wrong it was right not left

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.

Compare this function to the parent function f(x)
Y=f(x+1)

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 \( Y = f(x + 1) \).

Describe the transformation from the parent quadratic function
Y=(x-2)^2

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).

What change does a translation cause

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.

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

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:

\( Y = x^2 + 5 \).

Describe the transformation of
Y=x^2+4

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).

What transformation transforms the graph of f(x)=x^2 to the graph of g(x)=(x+4)^2

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).

What's the transformation of the graph y=(x-3)^3-2

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 \( (3, -2) \).