Unit 3 Parent Functions And Transformations Homework 2

Let's delve into the fascinating world of parent functions and transformations, a cornerstone of understanding function behavior and manipulation in mathematics. Mastering these concepts unlocks the ability to visualize and predict how altering a function's equation impacts its graph, opening doors to solving a wide range of problems across various disciplines.

Understanding Parent Functions

At their core, parent functions are the simplest form of a family of functions. They act as the blueprint, the foundational element upon which more complex functions are built through transformations. Knowing these parent functions intimately is key to quickly recognizing and analyzing more intricate equations. Here are some of the most common parent functions:

• Linear Function: f(x) = x
  • This is the most basic linear function, a straight line passing through the origin with a slope of 1. Its graph is a diagonal line that increases as x increases.
• Quadratic Function: f(x) = x²
  • The quadratic function forms a parabola, a U-shaped curve that opens upwards. The vertex of this parent function is located at the origin (0,0).
• Cubic Function: f(x) = x³
  • The cubic function has a distinctive S-shaped curve, passing through the origin. It increases more rapidly than the quadratic function as x moves away from zero.
• Square Root Function: f(x) = √x
  • Defined only for non-negative values of x, the square root function starts at the origin and increases gradually. Its graph forms a curve that becomes flatter as x increases.
• Absolute Value Function: f(x) = |x|
  • The absolute value function returns the magnitude of x, regardless of its sign. Its graph is a V-shape, with the vertex at the origin. For positive x, it's identical to the linear function, and for negative x, it's the reflection of the linear function across the x-axis.
• Reciprocal Function: f(x) = 1/x
  • Also known as a hyperbola, the reciprocal function has two distinct branches. It has vertical asymptote at x = 0 and a horizontal asymptote at y = 0.
• Exponential Function: f(x) = aˣ (where a > 0 and a ≠ 1)
  • Exponential functions demonstrate rapid growth or decay. The value of 'a' determines whether the function increases (a > 1) or decreases (0 < a < 1). The graph has a horizontal asymptote at y = 0.
• Logarithmic Function: f(x) = logₐ(x) (where a > 0 and a ≠ 1)
  • Logarithmic functions are the inverse of exponential functions. The value of 'a' is the base of the logarithm. The graph has a vertical asymptote at x = 0.

Types of Transformations

Transformations are operations that alter the position, shape, or size of a graph. Understanding how these transformations affect the parent function is crucial. They are broadly categorized into:

1. Vertical and Horizontal Shifts (Translations): These move the graph without changing its shape or orientation.
2. Vertical and Horizontal Stretches and Compressions (Dilations): These change the shape of the graph by stretching or compressing it along the x or y-axis.
3. Reflections: These flip the graph across the x-axis or y-axis.

Let's break down each type in detail:

1. Vertical and Horizontal Shifts (Translations)

• Vertical Shift: This transformation moves the graph up or down. If f(x) is the original function, then f(x) + k shifts the graph k units vertically.

  • If k > 0, the graph shifts upwards.
  • If k < 0, the graph shifts downwards.

  Example: Consider the parent function f(x) = x². If we add 3 to it, we get g(x) = x² + 3. This shifts the parabola 3 units upwards.

• Horizontal Shift: This transformation moves the graph left or right. If f(x) is the original function, then f(x - h) shifts the graph h units horizontally. Crucially, note the subtraction sign.

  • If h > 0, the graph shifts rightwards.
  • If h < 0, the graph shifts leftwards.

  Example: Consider the parent function f(x) = x². If we subtract 2 from x before squaring, we get g(x) = (x - 2)². This shifts the parabola 2 units to the right.

2. Vertical and Horizontal Stretches and Compressions (Dilations)

These transformations alter the 'spread' of the graph.

• Vertical Stretch or Compression: This transformation stretches or compresses the graph vertically. If f(x) is the original function, then a f(x) stretches or compresses the graph vertically by a factor of a.

  • If a > 1, the graph is vertically stretched (it becomes taller and narrower).
  • If 0 < a < 1, the graph is vertically compressed (it becomes shorter and wider).

  Example: Consider the parent function f(x) = x². If we multiply it by 2, we get g(x) = 2x². This vertically stretches the parabola, making it narrower. If we multiply it by 0.5, we get g(x) = 0.5x². This vertically compresses the parabola, making it wider.

• Horizontal Stretch or Compression: This transformation stretches or compresses the graph horizontally. If f(x) is the original function, then f(bx) stretches or compresses the graph horizontally by a factor of 1/b. Note the reciprocal relationship.

  • If b > 1, the graph is horizontally compressed (it becomes narrower).
  • If 0 < b < 1, the graph is horizontally stretched (it becomes wider).

  Example: Consider the parent function f(x) = x². If we substitute x with 2x, we get g(x) = (2x)² = 4x². This horizontally compresses the parabola by a factor of 1/2, making it narrower. If we substitute x with 0.5x, we get g(x) = (0.5x)² = 0.25x². This horizontally stretches the parabola by a factor of 2, making it wider.

3. Reflections

Reflections flip the graph across an axis.

• Reflection Across the x-axis: This flips the graph over the x-axis. If f(x) is the original function, then -f(x) reflects the graph across the x-axis. The y-values change sign.

  Example: Consider the parent function f(x) = x². If we multiply it by -1, we get g(x) = -x². This reflects the parabola across the x-axis, so it now opens downwards.

• Reflection Across the y-axis: This flips the graph over the y-axis. If f(x) is the original function, then f(-x) reflects the graph across the y-axis. The x-values change sign.

  Example: Consider the parent function f(x) = √x. If we replace x with -x, we get g(x) = √(-x). This reflects the graph across the y-axis. Note that the domain of the original function is x ≥ 0, while the domain of the transformed function is x ≤ 0. Important Note: Even functions (like x² or |x|) are unchanged by reflection across the y-axis because f(x) = f(-x).

Combining Transformations

Often, functions undergo multiple transformations simultaneously. The order in which these transformations are applied matters. A general form representing a transformed function is:

g(x) = a * f(b(x - h)) + k

Where:

• a represents a vertical stretch or compression (and reflection across the x-axis if negative).
• b represents a horizontal stretch or compression (and reflection across the y-axis if negative).
• h represents a horizontal shift.
• k represents a vertical shift.

The order of operations to apply these transformations to the parent function f(x) is crucial and follows the reverse of PEMDAS/BODMAS:

1. Horizontal Shift: Apply the horizontal shift (h) first. Replace x with (x - h) in the parent function.
2. Horizontal Stretch/Compression/Reflection: Apply the horizontal stretch or compression (b) and reflection across the y-axis (if b is negative). Replace (x - h) with b(x - h).
3. Vertical Stretch/Compression/Reflection: Apply the vertical stretch or compression (a) and reflection across the x-axis (if a is negative). Multiply the entire function by a.
4. Vertical Shift: Finally, apply the vertical shift (k). Add k to the entire function.

Example: Let's transform the parent function f(x) = x² using the equation g(x) = -2(x + 1)² - 3.

1. Horizontal Shift: h = -1, so we shift the parabola 1 unit to the left: (x + 1)²
2. Horizontal Stretch/Compression/Reflection: b = 1 (no horizontal change in this case).
3. Vertical Stretch/Compression/Reflection: a = -2, so we vertically stretch the parabola by a factor of 2 and reflect it across the x-axis: -2(x + 1)²
4. Vertical Shift: k = -3, so we shift the parabola 3 units down: -2(x + 1)² - 3

The resulting parabola is narrower, opens downwards, is shifted 1 unit to the left, and 3 units down.

Practical Applications and Examples

Understanding parent functions and transformations is vital for:

• Graphing Functions Quickly: By recognizing the parent function and the transformations applied, you can sketch the graph without plotting numerous points.
• Analyzing Function Behavior: Transformations reveal how changing parameters in an equation affect the function's key characteristics (e.g., vertex, intercepts, asymptotes).
• Modeling Real-World Phenomena: Many real-world situations can be modeled using transformed parent functions.
• Solving Equations: Transformations can simplify equations, making them easier to solve.

Let's look at some more detailed examples:

Example 1: Transforming the Square Root Function

Consider the function g(x) = -√(2(x - 3)) + 1. The parent function is f(x) = √x. Let's identify the transformations:

1. Horizontal Shift: The term (x - 3) indicates a horizontal shift of 3 units to the right.
2. Horizontal Compression: The 2 inside the square root, (2x), indicates a horizontal compression by a factor of 1/2.
3. Reflection Across the x-axis: The negative sign in front of the square root, -√(…), indicates a reflection across the x-axis.
4. Vertical Shift: The + 1 at the end indicates a vertical shift of 1 unit up.

To graph this, start with the parent function √x. Shift it 3 units right, compress it horizontally by a factor of 1/2, reflect it across the x-axis, and then shift it 1 unit up.

Example 2: Transforming the Absolute Value Function

Consider the function g(x) = 3|-(x + 2)| - 4. The parent function is f(x) = |x|. Let's break down the transformations:

1. Horizontal Shift: The term (x + 2) indicates a horizontal shift of 2 units to the left.
2. Reflection Across the y-axis: The negative sign inside the absolute value, |-x|, indicates a reflection across the y-axis. However, since the absolute value function is even (symmetric about the y-axis), this reflection doesn't visually change the graph at this stage. It's still important to acknowledge it in the transformation process.
3. Vertical Stretch: The 3 outside the absolute value, 3|…|, indicates a vertical stretch by a factor of 3.
4. Vertical Shift: The - 4 at the end indicates a vertical shift of 4 units down.

Starting with the parent function |x|, shift it 2 units left, (reflect across the y-axis - no visible change), stretch it vertically by a factor of 3, and then shift it 4 units down.

Example 3: Application - Modeling Projectile Motion

The height h(t) of a projectile launched vertically can be modeled by a quadratic function: h(t) = -at² + vt + s, where:

• t is time.
• a is related to the acceleration due to gravity.
• v is the initial vertical velocity.
• s is the initial height.

The parent function here is f(t) = t². The transformations are:

• Reflection across the t-axis (due to the negative sign in front of at²).
• Vertical stretch/compression by a factor of a.
• Horizontal and vertical shifts (determined by the values of v and s, and by completing the square to rewrite the quadratic in vertex form).

By understanding these transformations, you can analyze the projectile's trajectory, determine its maximum height, and predict when it will hit the ground.

Common Mistakes to Avoid

• Incorrect Order of Transformations: Applying transformations in the wrong order will lead to an incorrect graph. Remember to follow the reverse of PEMDAS/BODMAS.
• Misinterpreting Horizontal Shifts: Remember that f(x - h) shifts the graph h units to the right, not the left.
• Confusing Stretches and Compressions: Understand the reciprocal relationship in horizontal stretches and compressions.
• Ignoring Reflections: Don't forget to consider reflections across both the x and y-axes.
• Forgetting the Impact on Domain and Range: Transformations can affect the domain and range of a function. Be mindful of these changes, especially when dealing with square root, logarithmic, or rational functions.

Homework 2: Applying Your Knowledge

Now, let's consider what a typical "Unit 3 Parent Functions and Transformations Homework 2" might entail. It would likely involve problems that require you to:

1. Identify the Parent Function: Given a transformed function, recognize the underlying parent function.
2. Describe the Transformations: Given a transformed function, list all the transformations that have been applied to the parent function, in the correct order.
3. Write the Equation of a Transformed Function: Given a description of transformations, write the equation of the resulting function.
4. Sketch the Graph of a Transformed Function: Sketch the graph of a transformed function, using your knowledge of the parent function and the transformations.
5. Determine Key Features: Identify key features of the transformed function, such as its vertex, intercepts, asymptotes, domain, and range.

Example Homework Problems:

1. Identify the parent function and describe the transformations applied to obtain the function g(x) = -3(x + 2)² + 1.
2. Write the equation of the function that is obtained by shifting the square root function 4 units to the left, reflecting it across the x-axis, and stretching it vertically by a factor of 2.
3. Sketch the graph of the function h(x) = 2|x - 1| - 3. Identify the vertex and the range of the function.
4. The graph of f(x) = 1/x is transformed to obtain the graph of g(x) = 1/(x - 3) + 2. Describe the transformations and identify the equations of the asymptotes of g(x).
5. A projectile is launched with an initial height of 2 meters and an initial vertical velocity of 10 m/s. The height of the projectile is given by h(t) = -4.9t² + 10t + 2. Identify the parent function and describe the transformations. What is the maximum height reached by the projectile?

To effectively tackle such homework, a structured approach is beneficial:

• Step 1: Identify the Parent Function: Recognize the core function that has been transformed.
• Step 2: Analyze the Equation: Carefully examine the equation for values of a, b, h, and k. Pay close attention to signs.
• Step 3: List the Transformations: Write down each transformation, in the correct order.
• Step 4: Apply the Transformations (Graphically or Algebraically): Use your understanding of the transformations to sketch the graph or manipulate the equation as needed.
• Step 5: Verify Your Result: Check your answer to ensure it makes sense in the context of the problem.

Conclusion

Mastering parent functions and transformations is a fundamental skill in mathematics. By understanding the basic parent functions and the effects of various transformations, you can analyze, manipulate, and graph a wide range of functions with confidence. Practice is key to solidifying your understanding. Work through numerous examples, pay attention to the order of transformations, and don't be afraid to seek help when needed. With dedication and a solid grasp of the concepts, you'll unlock a powerful toolset for tackling mathematical challenges. This knowledge extends far beyond the classroom, providing a foundation for understanding and modeling phenomena in various fields of science, engineering, and beyond.