1.
Item 5. Determine the domain of the following graph.
Solution:The graph shows separate points at integer \(x\)-values.The arrows indicate that the pattern continues in both directions.Therefore, the domain is all integers: \(x\in\mathbb{Z}\).Answer: B
2.
Item 3. What is the maximum number of x-intercepts for a 6th-degree function?
Solution:The maximum number of \(x\)-intercepts of a polynomial is equal to its degree.A \(6\)th-degree function can have at most \(6\) x-intercepts.Answer: D
3.
Item 6. Which notation best represents the following line graph?
Solution:An open circle means the endpoint is not included.A round bracket means the endpoint is not included.Both \(-5\) and \(3\) are excluded, so the interval is \((-5,3)\).Answer: B
4.
Item 1. Which of the following lines represent the equation \(y=3x^2-2\)?
Solution:The equation \(y=3x^2-2\) is a quadratic graph that opens upward because \(a=3>0\).Its vertex is at \((0,-2)\), and it is vertically stretched because \(a=3\).On the graph, the parabola labelled \(E\) opens upward and has vertex \((0,-2)\).Answer: D
5.
Item 15. Which of the following statement is false for a horizontal reflection?
Solution:A horizontal reflection is a reflection across the \(y\)-axis.The invariant points must lie on the \(y\)-axis, so they have \(x=0\).These points are the \(y\)-intercepts, not the \(x\)-intercepts.Therefore, the false statement is that the invariant points are the x-intercepts.Answer: C
6.
Item 18. Prepare for transformation the following equation: \(y=-2(3x-12)^2-6\).
Solution:Start with:\[y=-2(3x-12)^2-6\]Factor the coefficient of \(x\) inside the brackets:\[3x-12=3(x-4)\]Substitute back:\[y=-2[3(x-4)]^2-6\]Answer: A
7.
Item 8. Solve \((+2)+(+8)\).
Solution:\[(+2)+(+8)=2+8=10\]Answer: A
8.
Item 2. Which type of graph is this?
Solution:The graph has opposite end behaviour and two turning points, which is typical of a cubic function.A cubic function has degree \(3\).Answer: C
9.
Item 19. The equation of \(y=\sqrt{x}\) is transformed to \(y=-4\sqrt{-3(x+1)}-2\). Give the appropriate order of transformations.
Solution:The standard order for transformations is:1. Reflections2. Expansions or compressions3. TranslationsIn \(y=-4\sqrt{-3(x+1)}-2\), the outside negative gives a vertical reflection, and the negative inside gives a horizontal reflection.The coefficient \(4\) gives a vertical expansion, and the coefficient \(3\) inside gives a horizontal compression.The \(x+1\) gives a horizontal translation left \(1\), and \(-2\) gives a vertical translation down \(2\).Answer: D
10.
Item 11. Write the equation of the graph after it has been compressed vertically by a factor of \(\frac{3}{7}\) from the original graph of \(y=(x-2)^3\).
Solution:A vertical compression by \(\frac{3}{7}\) means multiply the whole function by \(\frac{3}{7}\).Start with \(y=(x-2)^3\).\[y=\frac{3}{7}(x-2)^3\]Answer: A
11.
Item 7. Given the function \(y=f(x-3)\), determine the new equation after it has been translated \(5\) units right.
Solution:To translate a graph \(5\) units right, replace \(x\) with \(x-5\).Start with \(y=f(x-3)\).\[y=f((x-5)-3)\]\[y=f(x-8)\]Answer: B
12.
Item 17. If \((-6,1)\) is a point on the graph \(f(x)\), what point must be on the graph of \(y=2f(x)-3\)?
Solution:The transformation \(y=2f(x)-3\) changes the \(y\)-value only.Since \((-6,1)\) is on \(f(x)\), \(f(-6)=1\).\[y=2(1)-3=-1\]The \(x\)-value stays \(-6\).Therefore, the new point is \((-6,-1)\).Answer: C
13.
Item 10. If the point \(A(3,-1)\) is on the graph of \(y=f(x)\), which point \(A'\) must be on the transformed graph \(y=3f(x)\)?
Solution:The transformation \(y=3f(x)\) is a vertical expansion by a factor of \(3\).The \(x\)-value stays the same, and the \(y\)-value is multiplied by \(3\).\[(3,-1)\rightarrow(3,-1\cdot3)=(3,-3)\]Answer: A
14.
Item 20. Identify the order of transformations required to create graph B from graph A.
Solution:From graph A to graph B, the graph is first reflected horizontally.Then it is translated \(5\) units to the right.Therefore, the correct order is horizontal reflection, then horizontal translation \(5\) units right.Answer: A
15.
Item 4. State the domain of \(y=4x-1\).
Solution:The equation \(y=4x-1\) is a linear function.Linear functions have no restrictions on \(x\).Therefore, the domain is all real numbers: \(x\in\mathbb{R}\).Answer: A
16.
Item 13. Which of the following represents both a vertical reflection and a horizontal reflection? I. \(-6y=\sqrt{-x}\) II. \(-4y=\left(-\frac{2}{3}x\right)^3\) III. \(11y=-6^x\) IV. \(5y=(-2x-1)^2\) V. \(-5y=\frac{1}{2}|x+8|\) VI. \(-5y=1.1(4)^{-2x+7}\)
Solution:A vertical reflection occurs when \(y\) changes sign.A horizontal reflection occurs when \(x\) changes sign.I has \(-6y\) and \(\sqrt{-x}\), so it has both reflections.II has \(-4y\) and \(\left(-\frac{2}{3}x\right)^3\), so it has both reflections.VI has \(-5y\) and exponent \(-2x+7\), so it has both reflections.Therefore, the correct group is I, II and VI.Answer: C
17.
Item 16. If the original function is \(y=f(x)\), what transformation(s) occurred if the function is now \(y=f(-2x)\)?
Solution:In \(f(-2x)\), the negative sign causes a horizontal reflection across the \(y\)-axis.The factor \(2\) inside the function causes a horizontal compression by a factor of \(\frac{1}{2}\).Therefore, the transformations are a horizontal reflection and a horizontal compression.Answer: C
18.
Item 12. How is the graph of \(5y=2^x\) related to the graph of \(y=2^x\)?
Solution:Solve for \(y\):\[5y=2^x\]\[y=\frac{1}{5}2^x\]Multiplying the function by \(\frac{1}{5}\) is a vertical compression by \(\frac{1}{5}\).Answer: C
19.
Item 14. The point \(B(4,-7)\) is on a line on a graph. Which of the following points must be on its inverse?
Solution:For an inverse, interchange the \(x\)- and \(y\)-coordinates.\[(4,-7)\rightarrow(-7,4)\]Answer: D
20.
Item 9. If you replace \(x\) by \(x-3\) and \(y\) by \(y+2\) in the equation \(y=x\), the graph will:
Solution:Replacing \(x\) with \(x-3\) causes a horizontal shift \(3\) units right.Replacing \(y\) with \(y+2\) causes a vertical shift \(2\) units down.Therefore, the graph moves \(3\) right and \(2\) down.Answer: A
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