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AoPS Algebra 2 - Lesson 4, Functions

Critical Thinking Framework

Remind yourself of the problem solving strategy/approach recommendations before you tackle the problems.

Aditional Resources:

Problem 1

Problem Statement:

Given that \((3,15)\) is on the graph of \(y = f(x)\), find a point that must be on the graph of \(y = f(x) - 7\). Express your answer as an ordered pair \((a,b)\) where \(a\) and \(b\) are real numbers.

Deconstruction & Analysis:

  1. Givens:

    • The point \((3, 15)\) is on the graph of the function \( y = f(x) \).
    • You need to find a point on the graph of the new function \( y = f(x) - 7 \).

  2. Unknowns:

    • The new \(y\)-coordinate for the given \(x = 3\) after the transformation is applied.

Hints & Strategy Outline:

  • Hint 1: Understand the effect of subtracting a constant from a function. What happens to the entire graph of \( f(x) \) when you subtract 7 from it?

  • Hint 2: Consider the relationship between the point on the original function \( (3, 15) \) and how the new function \( y = f(x) - 7 \) modifies that \(y\)-coordinate.

  • Hint 3: Use the given point \((3, 15)\) and adjust only the \(y\)-coordinate according to the transformation \( f(x) - 7 \). The \(x\)-coordinate remains unchanged.

Problem 3

Problem Statement:

Given that \((2,-7)\) is on the graph of \(y = f(x)\), find a point that must be on the graph of \(y = f(x - 3)\). Express your answer as an ordered pair \((a,b)\) where \(a\) and \(b\) are real numbers.

Deconstruction & Analysis:

  1. Givens:

    • The point \((2, -7)\) is on the graph of \(y = f(x)\).
    • You are asked to find a point on the graph of \(y = f(x - 3)\).

  2. Unknowns:

  3. The new \(x\)-coordinate after the transformation of the function, while the \(y\)-coordinate remains unchanged.

Hints & Strategy Outline:

  • Hint 1: Understand the effect of a horizontal shift on the graph of a function. What does subtracting 3 inside the function do to the \(x\)-coordinates of the points on the graph?

  • Hint 2: For the given point \((2, -7)\), think about how the horizontal shift \(x - 3\) changes the original \(x\)-value. What new \(x\)-coordinate will result from this shift?

  • Hint 3: The \(y\)-coordinate remains the same because the transformation only affects the \(x\)-coordinate. Use this to determine the new point.

Problem 4

Problem Statement:

Given that \((-2,3)\) is on the graph of \(y=f(x)\), find a point that must be on the graph of \(y=f(2x+1)+3\). Express your answer as an ordered pair \((a,b)\) where \(a\) and \(b\) are real numbers.

Deconstruction & Analysis:

  1. Givens:

    • The point \((-2, 3)\) is on the graph of \(y = f(x)\).
    • You need to find a point on the graph of the transformed function \(y = f(2x + 1) + 3\).

  2. Unknowns:

    • The new \(x\)-coordinate after applying the transformation inside the function.
    • The new \(y\)-coordinate after considering both the transformation inside and the vertical shift outside the function.

Hints & Strategy Outline:

  • Hint 1: Start by understanding the inside transformation \(f(2x + 1)\). How does the \(2x + 1\) affect the \(x\)-coordinate of the original point? Consider solving for the original \(x\)-value that produces the same output in terms of \(f(x)\).

  • Hint 2: The vertical transformation \(+3\) affects the \(y\)-coordinate. After calculating the new \(y\)-value for the modified input, adjust it by adding 3 to reflect the vertical shift.

  • Hint 3: Use the original point \((-2, 3)\) to identify the required changes to both the \(x\)-coordinate (from the inside transformation) and the \(y\)-coordinate (from the outside transformation).

Problem 5

Let \(f(x)\) be the function graphed below. Need the visualization of graph, and in terms of \(f(x)\), what is the function graphed below?

Problem 6

Problem Statement:

The region between the graph of \(y = f (x)\) and the \(x\)-axis, shaded in this figure, has an area of 10 square units. What is the area between the graph of \(y = f (x +2)\) and the \(x\)-axis?

Deconstruction & Analysis:

  1. Givens:

    • The area between the graph of \(y = f(x)\) and the \(x\)-axis is 10 square units.
    • You need to determine the area between the graph of \(y = f(x + 2)\) and the \(x\)-axis.

  2. Unknowns:

    • The area between \(y = f(x + 2)\) and the \(x\)-axis, and how the horizontal shift affects the area.

Hints & Strategy Outline:

  • Hint 1: Consider how a horizontal shift, such as \(f(x + 2)\), impacts the shape and position of the graph of \(f(x)\). Does it change the overall size of the area between the graph and the \(x\)-axis, or does it merely shift it left or right?

  • Hint 2: Reflect on whether the transformation \(x + 2\) alters the actual integral (area) under the curve, or just the position of the curve itself along the \(x\)-axis.

  • Hint 3: Use the fact that the total area under a function is preserved under horizontal shifts. What does this tell you about how the area for \(y = f(x + 2)\) compares to the original area of 10 square units?

Problem 7

Problem Statement:

The region between the graph of \(y = f (x)\) and the \(x\)-axis, shaded in this figure, has an area of 10 square units. What is the area between the graph of \(y = 6f (x - 6)\) and the \(x\)-axis?

Deconstruction & Analysis:

  1. Givens:

    • The area between the graph of \(y = f(x)\) and the \(x\)-axis is 10 square units.
    • You are asked to find the area between the graph of \(y = 6f(x - 6)\) and the \(x\)-axis.

  2. Unknowns:

    • How the transformation \(6f(x - 6)\) affects the area between the graph and the \(x\)-axis.

Hints & Strategy Outline:

  • Hint 1: The transformation \(x - 6\) is a horizontal shift, which does not affect the area. Think about how the graph \(y = f(x)\) changes when shifted 6 units to the right.

  • Hint 2: The factor of 6 outside the function scales the entire graph vertically by 6. Consider how this scaling affects the area under the curve. Does scaling the function by 6 multiply the area by 6 as well?

  • Hint 3: Focus on the effect of vertical scaling on areas under a curve. The area of the region under the curve is proportional to the vertical scaling factor.

Conclusion:

The horizontal shift \(x - 6\) does not affect the area, but the vertical scaling by 6 does. Since the original area is 10 square units, multiplying the graph by 6 multiplies the area by 6.

Final Answer:

The area between the graph of \(y = 6f(x - 6)\) and the \(x\)-axis is \(10 \times 6 = 60\) square units.

Problem 8

Problem Statement:

The region between the graph of \(y = f (x)\) and the \(x\)-axis, shaded in this figure, has an area of 10 square units. What is the area between the graph of \(y = \dfrac 12 f (-x)\) and the \(x\)-axis?

Deconstruction & Analysis:

  1. Givens:

    • The area between the graph of \(y = f(x)\) and the \(x\)-axis is 10 square units.
    • You need to find the area between the graph of \(y = \frac{1}{2} f(-x)\) and the \(x\)-axis.

  2. Unknowns:

    • How the transformations \(\frac{1}{2}\) and \(f(-x)\) affect the area.

Hints & Strategy Outline:

  • Hint 1: The transformation \(f(-x)\) reflects the graph across the \(y\)-axis. Consider whether this reflection affects the area under the curve.

  • Hint 2: The factor of \(\frac{1}{2}\) scales the graph vertically. Think about how this scaling impacts the area under the curve.

  • Hint 3: The area of the region under a curve is proportional to the vertical scaling factor. If the original area is 10 square units, how will the area change when you multiply the function by \(\frac{1}{2}\)?

Conclusion:

The reflection does not change the area, but the vertical scaling by \(\frac{1}{2}\) reduces the area.

Problem 9

Problem Statement:

When the graph of \(y = 2x^2 - x + 7\) is shifted four units to the right, we obtain the graph of \(y = ax^2 + bx + c\). Find \(a + b + c\).

Let's deconstruct the problem step by step.

Givens:

  1. The original function is \( y = 2x^2 - x + 7 \).
  2. The graph is shifted four units to the right.

Unknowns:

  1. The coefficients \(a\), \(b\), and \(c\) of the new function \(y = ax^2 + bx + c\) after the shift.
  2. The sum \(a + b + c\).

Conditions/Constraints:

  1. The transformation involves shifting the graph horizontally, which affects how the variable \(x\) is represented in the function.
  2. The new function's form remains quadratic, \(y = ax^2 + bx + c\).

Hints for Strategy:

  1. Understand the Transformation: Focus on how a horizontal shift affects the function. Recall that shifting the graph right involves replacing \(x\) with \(x - h\), where \(h\) is the shift amount.
  2. Substitution: Apply the transformation to the entire function, substituting \(x - 4\) into the original function.
  3. Expand and Simplify: After substitution, ensure to expand and combine like terms carefully to express the new function in the standard quadratic form.
  4. Identify Coefficients: Once you have the new function in the form \(y = ax^2 + bx + c\), clearly identify the coefficients \(a\), \(b\), and \(c\).
  5. Calculate the Sum: Finally, compute the sum \(a + b + c\) using the identified coefficients.

By focusing on these elements, you can systematically approach the problem and arrive at the solution without rushing into calculations.

Problem 10

Problem Statement:

The function \(f(x)\) is invertible, but the function \(g(x)=f(kx)\) is not invertible. Find the sum of all possible values of \(k\).

To analyze the problem, we start with the given functions and the conditions of invertibility.

Givens:

  1. The function \(f(x)\) is invertible. This means that \(f(x)\) is one-to-one, which implies it is either strictly increasing or strictly decreasing over its domain.
  2. The function \(g(x) = f(kx)\) is not invertible.

Unknowns:

  1. We need to find the sum of all possible values of \(k\) for which \(g(x)\) is not invertible.

Conditions/Constraints:

  1. A function is not invertible if it fails the horizontal line test, which occurs when it is not one-to-one.
  2. For \(g(x) = f(kx)\) to be non-invertible, the composition \(f(kx)\) must have the same output for different inputs.

Deconstruction:

  1. Function \(g(x)\): The expression \(g(x) = f(kx)\) suggests that we are transforming the input to \(f\) by multiplying \(x\) by \(k\).
  2. Behavior Based on \(k\):
  3. If \(k = 0\), then \(g(x) = f(0)\) is a constant function (which is not invertible).
  4. If \(k > 0\) or \(k < 0\), the behavior of \(g(x)\) depends on whether it stretches or compresses the input values into the domain of \(f\).
  5. Analyzing Non-Invertibility:
  6. The function \(g(x)\) could become non-invertible if \(kx\) takes on the same values for different \(x\). This can happen when \(k\) is such that it maps multiple \(x\) values to the same \(kx\) value.

Hints for Strategy:

  1. Investigate \(k\): Start by setting up conditions under which \(kx_1 = kx_2\) for \(x_1 \neq x_2\). This typically occurs if \(k = 0\).
  2. Consider the cases: Analyze the case for \(k > 0\) and \(k < 0\) to determine whether they can lead to non-invertibility. Focus on how the multiplication by \(k\) changes the input domain.
  3. Use derivative tests: If \(f'(x) = 0\) at some point, \(f(kx)\) could also become non-invertible depending on \(k\) and where \(kx\) lands relative to the points where \(f'(x) = 0\).
  4. Summation of \(k\) values: Once the values of \(k\) are identified, compute their sum.

By following these steps, you can systematically deduce the values of \(k\) for which \(g(x)\) is not invertible.

Problem 11

Problem Statement:

When the graph of a certain function \(f(x)\) is shifted \(2\) units to the right and stretched vertically by a factor of \(2\) relative to the \(x\)-axis (meaning that all \(y\)-coordinates are doubled), the resulting graph is identical to the original graph.

Given that \(f(0)=1\), what is \(f(10)\)?

Step-wise Solutions

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