Q1: What is the value of \(|-7|\)? (a) -7 (b) 7 (c) 0 (d) 14
Solution:
Ans: (b) Explanation: The absolute value of a number is its distance from zero on the number line, which is always non-negative. Therefore, \(|-7| = 7\).
Q2: Which of the following represents the graph of \(y = |x|\)? (a) A straight line passing through the origin (b) A V-shaped graph with vertex at the origin (c) A parabola opening upward (d) A horizontal line
Solution:
Ans: (b) Explanation: The graph of \(y = |x|\) is a V-shaped graph with its vertex at the origin. For positive \(x\), \(y = x\), and for negative \(x\), \(y = -x\), creating the characteristic V-shape.
Q3: Solve the equation \(|x| = 5\). (a) \(x = 5\) only (b) \(x = -5\) only (c) \(x = 5\) or \(x = -5\) (d) No solution
Solution:
Ans: (c) Explanation: When solving absolute value equations, we consider both the positive and negative cases. Since the distance from zero can be 5 in either direction, both \(x = 5\) and \(x = -5\) are solutions.
Q4: For the piecewise function \(f(x) = \begin{cases} x + 3 & \text{if } x < 2="" \\="" 2x="" -="" 1="" &="" \text{if="" }="" x="" \geq="" 2="" \end{cases}\),="" what="" is=""> (a) 5 (b) 3 (c) 4 (d) 1
Solution:
Ans: (b) Explanation: Since \(x = 2\) and the condition \(x \geq 2\) is satisfied, we use the second piece: \(f(2) = 2(2) - 1 = 4 - 1 = 3\).
Q5: Which inequality represents all values of \(x\) such that \(|x - 3| <> (a) \(-1 < x=""><> (b) \(x <> (c) \(x > -1\) (d) \(-7 < x=""><>
Q6: What is the vertex of the absolute value function \(y = |x - 2| + 5\)? (a) \((-2, 5)\) (b) \((2, 5)\) (c) \((2, -5)\) (d) \((-2, -5)\)
Solution:
Ans: (b) Explanation: For the function \(y = |x - h| + k\), the vertex is at the point \((h, k)\). Here, \(h = 2\) and \(k = 5\), so the vertex is \((2, 5)\).
Q7: Evaluate the piecewise function \(g(x) = \begin{cases} x^2 & \text{if } x \leq 1 \\ 3x & \text{if } x > 1 \end{cases}\) at \(x = -2\). (a) -4 (b) 4 (c) -6 (d) 6
Solution:
Ans: (b) Explanation: Since \(x = -2\) and \(-2 \leq 1\), we use the first piece of the function: \(g(-2) = (-2)^2 = 4\).
Q8: How many solutions does the equation \(|2x + 1| = -3\) have? (a) 0 (b) 1 (c) 2 (d) Infinitely many
Solution:
Ans: (a) Explanation: An absolute value is always non-negative, so it can never equal a negative number. Therefore, \(|2x + 1| = -3\) has no solution.
Section B: Fill in the Blanks
Q9:The absolute value of a number represents its __________ from zero on the number line.
Solution:
Ans: distance Explanation:Absolute value measures how far a number is from zero, regardless of direction, which is the definition of distance on the number line.
Q10:The vertex form of an absolute value function is \(y = a|x - h| + k\), where the vertex is at the point __________.
Solution:
Ans: \((h, k)\) Explanation: In the vertex form of an absolute value function, the values \(h\) and \(k\) represent the coordinates of the vertex.
Q11:For the equation \(|x| = 8\), the two solutions are \(x = 8\) and \(x =\) __________.
Solution:
Ans: -8 Explanation: Absolute value equations have two solutions when the right side is positive. Both 8 and -8 are 8 units away from zero, so both are solutions.
Q12:A __________ function is defined by different expressions over different intervals of its domain.
Solution:
Ans: piecewise Explanation: A piecewise function uses different formulas or rules for different parts of its domain, typically separated by conditions or intervals.
Q13:The solution to the inequality \(|x| > 5\) is \(x < -5\)="" or="" \(x="">\) __________.
Solution:
Ans: 5 Explanation: When \(|x| > 5\), the number \(x\) must be more than 5 units away from zero in either direction, giving \(x < -5\)="" or="" \(x=""> 5\).
Q14:If \(f(x) = |x + 4|\), then \(f(-4) =\) __________.
Solution:
Ans: 0 Explanation: Substituting \(x = -4\) into the function: \(f(-4) = |-4 + 4| = |0| = 0\).
Section C: Word Problems
Q15:The temperature in a city varies throughout the day. The temperature \(T\) in degrees Fahrenheit at hour \(h\) after midnight is given by the piecewise function:
\[T(h) = \begin{cases} 50 + 2h & \text{if } 0 \leq h < 6="" \\="" 62="" &="" \text{if="" }="" 6="" \leq="" h="">< 18="" \\="" 80="" -="" h="" &="" \text{if="" }="" 18="" \leq="" h="" \leq="" 24="" \end{cases}\]="" what="" is="" the="" temperature="" at="" 4="">
Q16:A delivery company charges for packages based on weight. The cost \(C\) in dollars for a package weighing \(w\) pounds is:
\[C(w) = \begin{cases} 5 & \text{if } 0 < w="" \leq="" 2="" \\="" 5="" +="" 3(w="" -="" 2)="" &="" \text{if="" }="" w=""> 2 \end{cases}\]
How much does it cost to ship a package weighing 7 pounds?
Solution:
Ans: Since \(w = 7\) and \(7 > 2\), we use the second piece: \(C(7) = 5 + 3(7 - 2)\) \(C(7) = 5 + 3(5)\) \(C(7) = 5 + 15\) \(C(7) = 20\) Final Answer: $20
Q17:Maria lives 8 blocks from school. She walks to school and then walks home. The distance \(d\) she is from home after walking \(x\) blocks can be modeled by \(d = |x - 8|\). How far is Maria from home after walking 5 blocks?
Solution:
Ans: Using the function \(d = |x - 8|\) with \(x = 5\): \(d = |5 - 8|\) \(d = |-3|\) \(d = 3\) Final Answer: 3 blocks
Q18:Solve the equation \(|3x - 6| = 12\) to find all possible values of \(x\).
Solution:
Ans: For an absolute value equation \(|A| = B\) where \(B > 0\), we have \(A = B\) or \(A = -B\).
Case 1: \(3x - 6 = 12\) \(3x = 18\) \(x = 6\)
Case 2: \(3x - 6 = -12\) \(3x = -6\) \(x = -2\) Final Answer: \(x = 6\) or \(x = -2\)
Q19:The ideal operating temperature for a machine is 75°F. The machine will function properly if the actual temperature \(T\) differs from the ideal by no more than 10°F. Write and solve an absolute value inequality to find the range of acceptable temperatures.
Solution:
Ans: The difference from the ideal temperature must be at most 10°F: \(|T - 75| \leq 10\)
This translates to: \(-10 \leq T - 75 \leq 10\)
Adding 75 to all parts: \(65 \leq T \leq 85\) Final Answer: The acceptable temperature range is 65°F to 85°F
Q20:A parking garage charges based on the number of hours parked. For the first 2 hours, the cost is $3 per hour. For any time over 2 hours, the cost is $5 per hour for the additional time. Write a piecewise function \(C(h)\) for the total cost where \(h\) is the number of hours parked, and find the cost for parking 5 hours.
Solution:
Ans: The piecewise function is:
\[C(h) = \begin{cases} 3h & \text{if } 0 < h="" \leq="" 2="" \\="" 6="" +="" 5(h="" -="" 2)="" &="" \text{if="" }="" h=""> 2 \end{cases}\]
For \(h = 5\), since \(5 > 2\), we use the second piece: \(C(5) = 6 + 5(5 - 2)\) \(C(5) = 6 + 5(3)\) \(C(5) = 6 + 15\) \(C(5) = 21\) Final Answer: $21
The document Worksheet (with Solutions): Absolute Value & Piecewise Functions is a part of the Grade 9 Course Mathematics: Algebra 1.
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