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Question 1. On the graph shown, what is \(f(-2)\)?
Step 1. Locate \(x=-2\) on the graph. Step 2. At \(x=-2\), the graph has a closed point at \(y=3\). Open circles are not included, but the closed point gives the function value. Step 3. Therefore, \(f(-2)=3\). Answer: A. \(3\)
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Question 2. Which of the following graphs represents a function?
Step 1. Use the vertical line test. Step 2. A graph represents a function if every vertical line crosses the graph at no more than one point. Step 3. Graph A, Graph B, and Graph C fail the vertical line test. Step 4. Graph D passes the vertical line test because each \(x\)-value has only one \(y\)-value. Answer: D. Graph D
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Question 3. Determine the domain of the relation graphed below.
Step 1. The domain is the set of all possible \(x\)-values on the graph. Step 2. The graph begins at \(x=1\) with a closed point, so \(1\) is included. Step 3. The graph ends at \(x=3\) with an open point, so \(3\) is not included. Step 4. Therefore, the domain is \([1,3)\). Answer: C. \([1,3)\)
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Question 4. State the range of the function.
Step 1. The range is the set of all possible \(y\)-values on the graph. Step 2. The highest point on the graph has \(y=3\), and it is included. Step 3. The lowest point on the graph has \(y=-5\), and it is included. Step 4. Therefore, the range is \(-5\le y\le 3\). Answer: B. \(-5\le y\le 3\)
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Question 5. How is the graph of \(y=(x-3)^3-5\) related to the graph of \(y=(x-3)^3\)?
Step 1. Compare \(y=(x-3)^3\) with \(y=(x-3)^3-5\). Step 2. Subtracting \(5\) outside the function moves every point vertically downward by \(5\) units. Step 3. Therefore, the graph has been translated \(5\) units down. Answer: C. It has been translated 5 units down.
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Question 6. The graph of \(y=f(x)\) is transformed to \(y=f(x-p)+q\) to move the graph 3 units left and 3 units down. Find the value of \(p\) and \(q\).
Step 1. In \(y=f(x-p)+q\), the value \(p\) controls the horizontal translation. Step 2. To move 3 units left, use \(x+3\), which means \(x-p=x+3\), so \(p=-3\). Step 3. The value \(q\) controls the vertical translation. Step 4. To move 3 units down, \(q=-3\). Answer: C. \(p=-3, q=-3\)
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Question 7. The function defined as \(T(x)=f(x+3)\) is a transformation of the function \(f(x)\) shown below. If point \(A(1,-2)\) is on the graph of \(f(x)\), what will be the coordinates of \(A'\) on the transformed function \(T(x)\)?
Step 1. The transformation \(T(x)=f(x+3)\) shifts the graph of \(f(x)\) left by \(3\) units. Step 2. A horizontal shift left by \(3\) changes the \(x\)-coordinate from \(1\) to \(1-3=-2\). Step 3. The \(y\)-coordinate does not change, so it remains \(-2\). Step 4. Therefore, \(A'(-2,-2)\). Answer: D. \(A'(-2,-2)\)
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Question 8. Given \(y=f(x)\), which of the following changes will reflect a graph in the \(x\)-axis?
Step 1. A reflection in the \(x\)-axis changes every point \((x,y)\) into \((x,-y)\). Step 2. This means the \(y\)-values are multiplied by \(-1\). Step 3. Replacing \(y\) by \(-y\) produces a reflection in the \(x\)-axis. Answer: C. \(y\) is replaced by \(-y\)
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Question 9. If you were to reflect the graph of \(y=\frac{1}{2}x^3-1\) in the \(x\)-axis, its equation would be:
Step 1. A reflection in the \(x\)-axis multiplies the entire function by \(-1\). Step 2. Start with \(y=\frac{1}{2}x^3-1\). Step 3. Multiply by \(-1\): \(y=-\left(\frac{1}{2}x^3-1\right)\). Step 4. Simplify: \(y=-\frac{1}{2}x^3+1\). Answer: B. \(y=-\frac{1}{2}x^3+1\)
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Question 10. The function \(y=g(x)\) is shown in Diagram 1. The equation of the function shown in Diagram 2 could be:
Step 1. Compare Diagram 1 and Diagram 2. Step 2. Diagram 2 has the same horizontal positions as Diagram 1, but the \(y\)-values are opposite in sign. Step 3. This is a reflection in the \(x\)-axis. Step 4. A reflection in the \(x\)-axis is represented by \(y=-g(x)\). Answer: B. \(y=-g(x)\)
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Question 11. If the graph of the equation \(y=x+5\) were compressed horizontally by a factor of \(\frac{1}{2}\) and reflected in the \(y\)-axis, the new equation would be:
Step 1. Let \(f(x)=x+5\). Step 2. A horizontal compression by a factor of \(\frac{1}{2}\) means replace \(x\) by \(2x\), giving \(f(2x)=2x+5\). Step 3. Reflecting in the \(y\)-axis means replace \(x\) by \(-x\), so use \(f(-2x)\). Step 4. Calculate \(f(-2x)=-2x+5\). Answer: D. \(y=-2x+5\)
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Question 12. The graph of \(y=f(x)\) is transformed by a horizontal compression by a factor of \(\frac{1}{4}\) and a vertical expansion by a factor of \(5\). The equation of this new image has the form \(y=af(bx)\); find \(a\) and \(b\).
Step 1. In \(y=af(bx)\), \(a\) controls the vertical stretch or compression. Step 2. A vertical expansion by a factor of \(5\) gives \(a=5\). Step 3. A horizontal compression by a factor of \(\frac{1}{4}\) means the input is multiplied by \(4\), so \(b=4\). Step 4. Therefore, \(a=5\) and \(b=4\). Answer: D. \(a=5, b=4\)
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Question 13. The function \(T(x)\) is a transformation of the function \(f(x)\). Which transformation(s) on \(f(x)\) are required to obtain the function \(T(x)\)?
Step 1. Compare the direction of opening of \(f(x)\) and \(T(x)\). Step 2. Since \(f(x)\) opens upward and \(T(x)\) opens downward, the graph is reflected in the \(x\)-axis. Step 3. Compare the vertex positions on the graph. Step 4. The vertex moves 4 units left and 3 units up. Step 5. Therefore, the required transformations are reflection in the \(x\)-axis, vertical translation 3 units up, and horizontal translation 4 units left. Answer: C. \(f(x)\) is reflected in the \(x\)-axis, vertically translated 3 units up, and horizontally translated 4 units to the left.
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Question 14. How is the graph \(y=\left(\frac{1}{3}(x+2)\right)^2\) related to the graph of \(y=x^2\)?
Step 1. Rewrite the function as \(y=\left(\frac{x+2}{3}\right)^2\). Step 2. The factor \(\frac{1}{3}\) inside the input causes a horizontal stretch by a factor of \(3\). Step 3. The expression \(x+2\) shifts the graph 2 units left. Step 4. Therefore, the graph has a horizontal stretch by \(3\) and a translation 2 units left. Answer: A. 1) horizontal stretch by a factor of 3 and 3) translation of 2 units left.
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Question 15. The function \(f(x)\) contains the point \(P(2,4)\). If the function \(f(x)\) is transformed into the function \(g(x)=-2f(2x)-1\), then point \(P\) will be transformed to the point \(P'(x,y)\). The values of \(x\) and \(y\), respectively, are:
Step 1. The original point is \(P(2,4)\), so the input is \(2\) and the output is \(4\). Step 2. In \(g(x)=-2f(2x)-1\), the inside \(2x\) means the \(x\)-coordinate is divided by \(2\): \(x'=\frac{2}{2}=1\). Step 3. The outside transformation changes the \(y\)-value by multiplying by \(-2\) and subtracting \(1\). Step 4. Calculate \(y'=-2(4)-1=-8-1=-9\). Step 5. Therefore, \(P'=(1,-9)\). Answer: D. 1 and -9
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Question 16. Which of the following equations would produce a graph with the general shape and position shown below?
Step 1. Identify the parent function as a square-root function. Step 2. The graph shown opens to the right, so the expression under the radical should be \(x\), not \(2-x\). Step 3. The graph is reflected downward, so the coefficient outside the radical is negative. Step 4. The graph is translated 2 units down, so subtract \(2\). Step 5. Therefore, the equation is \(y=-\sqrt{x}-2\). Answer: D. \(y=-\sqrt{x}-2\)
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Question 17. The domain of the function \(y=\sqrt{-3-4x}+2\) is \(x\le k\). The value of \(k\) is:
Step 1. For a square-root function, the expression inside the radical must be greater than or equal to \(0\). Step 2. Set the radicand greater than or equal to \(0\): \(-3-4x\ge 0\). Step 3. Add \(3\) to both sides: \(-4x\ge 3\). Step 4. Divide by \(-4\). Since dividing by a negative reverses the inequality, \(x\le -\frac{3}{4}\). Step 5. Convert \(-\frac{3}{4}\) to decimal form: \(-0.75\). Step 6. Therefore, \(k=-0.75\). Answer: D. \(-0.75\)
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Question 18. How is the graph of \(y=4|x|\) related to the graph of \(y=|x|\)?
Step 1. Compare \(y=|x|\) with \(y=4|x|\). Step 2. The multiplier \(4\) is outside the absolute value function. Step 3. A multiplier outside the function changes the \(y\)-values, causing a vertical stretch. Step 4. Since every \(y\)-value is multiplied by \(4\), the graph is stretched vertically by a factor of \(4\). Answer: C. The graph of \(y=|x|\) has been stretched vertically by a factor of \(4\) about the \(x\)-axis.
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