Q1: Which transformation shifts the graph of \(f(x) = x^2\) to the left 3 units? (a) \(f(x) = (x - 3)^2\) (b) \(f(x) = (x + 3)^2\) (c) \(f(x) = x^2 - 3\) (d) \(f(x) = x^2 + 3\)
Solution:
Ans: (b) Explanation: A horizontal shift to the left by \(h\) units is represented by replacing \(x\) with \((x + h)\). Therefore, shifting left 3 units gives \(f(x) = (x + 3)^2\). Option (a) shifts right, and options (c) and (d) are vertical shifts.
Q2: The function \(g(x) = -2f(x)\) represents which transformations of \(f(x)\)? (a) Vertical stretch by 2 only (b) Reflection over the x-axis and vertical stretch by 2 (c) Reflection over the y-axis and vertical stretch by 2 (d) Horizontal compression by 2
Solution:
Ans: (b) Explanation: The negative sign indicates a reflection over the x-axis, and the coefficient 2 represents a vertical stretch by a factor of 2. These transformations are applied to the output values of the function.
Q3: If \(f(x) = \sqrt{x}\), what is the equation of the function after shifting 4 units down and 2 units right? (a) \(g(x) = \sqrt{x - 2} - 4\) (b) \(g(x) = \sqrt{x + 2} - 4\) (c) \(g(x) = \sqrt{x - 4} - 2\) (d) \(g(x) = \sqrt{x + 4} + 2\)
Solution:
Ans: (a) Explanation: A horizontal shift right 2 units replaces \(x\) with \((x - 2)\), and a vertical shift down 4 units subtracts 4 from the function. Thus, \(g(x) = \sqrt{x - 2} - 4\).
Q4: What type of transformation does \(h(x) = f(3x)\) represent? (a) Vertical compression by a factor of 3 (b) Horizontal compression by a factor of 3 (c) Horizontal stretch by a factor of 3 (d) Vertical stretch by a factor of 3
Solution:
Ans: (b) Explanation: When the input is multiplied by a constant \(a > 1\), the graph undergoes a horizontal compression by a factor of \(a\). Here, \(f(3x)\) compresses the graph horizontally by a factor of 3.
Q5: The graph of \(y = |x|\) is reflected over the y-axis. What is the resulting function? (a) \(y = -|x|\) (b) \(y = |x|\) (c) \(y = |-x|\) (d) \(y = |x + 1|\)
Solution:
Ans: (b) Explanation: Reflecting over the y-axis replaces \(x\) with \(-x\), giving \(y = |-x|\). However, since \(|-x| = |x|\) for all \(x\), the function remains unchanged. The absolute value function is symmetric about the y-axis.
Q6: Which function represents a vertical stretch of \(f(x) = x^3\) by a factor of 5? (a) \(g(x) = (5x)^3\) (b) \(g(x) = 5x^3\) (c) \(g(x) = x^3 + 5\) (d) \(g(x) = (x/5)^3\)
Solution:
Ans: (b) Explanation: A vertical stretch by a factor of 5 multiplies the entire function by 5, giving \(g(x) = 5f(x) = 5x^3\). Option (a) represents a horizontal compression.
Q7: The transformation \(f(x) = (x - 5)^2 + 3\) represents which shifts from the parent function \(y = x^2\)? (a) Left 5, up 3 (b) Right 5, down 3 (c) Right 5, up 3 (d) Left 5, down 3
Solution:
Ans: (c) Explanation: The form \((x - h)^2 + k\) represents a horizontal shift right by \(h\) and a vertical shift up by \(k\). Here, \(h = 5\) and \(k = 3\), so the graph shifts right 5 units and up 3 units.
Q8: If \(g(x) = f(-x)\), which transformation is applied to \(f(x)\)? (a) Reflection over the x-axis (b) Reflection over the y-axis (c) Horizontal shift right (d) Vertical shift down
Solution:
Ans: (b) Explanation: Replacing \(x\) with \(-x\) causes a reflection over the y-axis. This transformation flips the graph horizontally across the vertical axis.
Section B: Fill in the Blanks
Q9:A transformation that flips a graph over a line is called a __________.
Solution:
Ans: reflection Explanation: A reflection is a transformation that produces a mirror image of the graph across a specified line, such as the x-axis or y-axis.
Q10:In the function \(f(x) = a(x - h)^2 + k\), the value of \(h\) determines the __________ shift.
Solution:
Ans: horizontal Explanation: The parameter \(h\) in the vertex form of a quadratic function controls the horizontal shift of the parabola.
Q11:The transformation \(g(x) = f(x) - 7\) shifts the graph of \(f(x)\) down by __________ units.
Solution:
Ans: 7 Explanation: Subtracting a constant from a function results in a vertical shift downward by that constant. Here, the graph shifts down 7 units.
Q12:If \(|a| > 1\) in \(g(x) = a \cdot f(x)\), the graph undergoes a vertical __________.
Solution:
Ans: stretch Explanation: When the coefficient \(a\) has an absolute value greater than 1, the graph experiences a vertical stretch, making it narrower or steeper.
Q13:The function \(h(x) = f(x + 9)\) represents a horizontal shift to the __________ by 9 units.
Solution:
Ans: left Explanation: Adding a positive constant inside the function argument shifts the graph horizontally to the left. The transformation \(f(x + 9)\) moves the graph 9 units left.
Ans: compression Explanation: A coefficient between 0 and 1 compresses the graph vertically, making it wider or less steep. This is a vertical compression.
Section C: Word Problems
Q15:The parent function \(f(x) = x^2\) is transformed to \(g(x) = 3(x + 2)^2 - 5\). Describe all transformations applied to the parent function and determine the vertex of \(g(x)\).
Solution:
Ans: The transformations applied are: 1. Horizontal shift left 2 units: \((x + 2)\) 2. Vertical stretch by a factor of 3: coefficient 3 3. Vertical shift down 5 units: \(-5\)
The vertex form of a parabola is \(g(x) = a(x - h)^2 + k\) where the vertex is \((h, k)\). Rewriting: \(g(x) = 3(x - (-2))^2 + (-5)\) Therefore, \(h = -2\) and \(k = -5\). Final Answer: The vertex is \((-2, -5)\).
Q16:A software engineer designs a parabolic antenna. The original signal function is \(f(x) = x^2\). To optimize reception, the antenna is stretched vertically by a factor of 4 and shifted up 6 units. Write the equation of the transformed function.
Solution:
Ans: Start with the parent function: \(f(x) = x^2\) Vertical stretch by 4: Multiply the function by 4: \(4x^2\) Vertical shift up 6 units: Add 6: \(4x^2 + 6\) Final Answer: \(g(x) = 4x^2 + 6\)
Q17:The function modeling a company's profit is \(P(t) = 200\sqrt{t}\), where \(t\) is time in months and \(P\) is profit in thousands of dollars. Due to a market change, the profit function is transformed to \(Q(t) = 200\sqrt{t - 3} + 50\). How many months later does the new profit start, and what is the initial profit increase?
Horizontal shift: \(t - 3\) means the function shifts right 3 units. This indicates profit starts 3 months later.
Vertical shift: \(+50\) means the function shifts up 50 units. Since profit is measured in thousands of dollars, this represents an initial increase of 50 thousand dollars or $50,000.
Final Answer: The new profit starts 3 months later with an initial profit increase of $50,000.
Q18:A biologist models bacterial growth with \(f(x) = 2^x\), where \(x\) is time in hours. The model is adjusted to \(g(x) = -2^x + 100\) to account for a limiting nutrient. Describe the transformations and explain what the value 100 represents in this context.
Reflection over the x-axis: The negative sign reflects the graph, making growth values negative. Vertical shift up 100 units: Adding 100 shifts the entire graph upward.
In context, the reflection and shift model a decreasing function starting from 100. The value 100 represents the carrying capacity or maximum sustainable population. The bacteria start at this limit and decrease over time.
Final Answer: The transformations are a reflection over the x-axis and a vertical shift up 100 units. The value 100 represents the carrying capacity of the bacterial population.
Q19:The graph of \(f(x) = |x|\) is transformed by reflecting it over the x-axis, then shifting it right 4 units and up 2 units. Write the equation of the transformed function and find the vertex.
Solution:
Ans: Start with \(f(x) = |x|\).
Reflection over x-axis: \(-|x|\) Horizontal shift right 4 units: Replace \(x\) with \((x - 4)\): \(-|x - 4|\) Vertical shift up 2 units: Add 2: \(-|x - 4| + 2\)
The transformed function is \(g(x) = -|x - 4| + 2\).
The vertex of \(|x - h| + k\) is at \((h, k)\). For \(g(x) = -|x - 4| + 2\), the vertex is at \((4, 2)\).
Final Answer: \(g(x) = -|x - 4| + 2\); vertex is \((4, 2)\).
Q20:A suspension bridge's cable follows the function \(f(x) = 0.05x^2\). Engineers redesign the bridge so the cable's function becomes \(g(x) = 0.05(2x)^2 + 10\). By what factor is the graph compressed horizontally, and how many units is it shifted vertically?
However, to identify the horizontal compression, examine the input transformation: The term \((2x)\) indicates the input is multiplied by 2, which causes a horizontal compression by a factor of 2.
The \(+10\) indicates a vertical shift up 10 units.
Final Answer: The graph is compressed horizontally by a factor of 2 and shifted up 10 units.
The document Worksheet (with Solutions): Transformations Of Functions is a part of the Grade 9 Course Mathematics: Algebra 2.
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