1.
Item 1. Which of the following lines represents the equation \(y=-2x-1\)?
Solution:The equation \(y=-2x-1\) has slope \(-2\) and \(y\)-intercept \(-1\).From the graph, line \(D\) has slope \(-2\) and crosses the \(y\)-axis at \(-1\).Therefore, the correct line is \(D\).Answer: B
2.
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 creates a horizontal reflection across the \(y\)-axis.The factor \(2\) creates a horizontal compression by a factor of \(\frac{1}{2}\).Therefore, the transformations are horizontal reflection and horizontal compression.Answer: B
3.
Item 2. How many possible solutions does a quadratic equation have?
Solution:A quadratic equation has degree \(2\).The maximum number of possible solutions is equal to the degree.Therefore, a quadratic equation can have up to \(2\) solutions.Answer: A
4.
Item 8. Solve: \((-5)-(-9)\).
Solution:\[(-5)-(-9)=-5+9=4\]Answer: C
5.
Item 5. This table shows the cost, \(C\) dollars, of different numbers of tickets sold, \(n\). Identify the range.
Solution:The range is the set of output values.In the table, the output values are the costs \(C\).The costs are \(12.50,25.00,37.50,50.00,62.50,\ldots\).Therefore, the range is \(\{12.5,25.00,37.50,50.00,62.50,\ldots\}\).Answer: D
6.
Item 17. What happens to the graph of a function if you replace \(x\) with \(5x\) and \(y\) with \(5y\)?
Solution:Replacing \(x\) with \(5x\) causes a horizontal compression by \(\frac{1}{5}\).Replacing \(y\) with \(5y\) causes a vertical compression by \(\frac{1}{5}\).Therefore, both directions are compressed by \(\frac{1}{5}\).Answer: B
7.
Item 13. Which transformation is represented in the graph?
Solution:The graph represents an ellipse reflected across the \(x\)-axis.A reflection across the \(x\)-axis changes the sign of the \(y\)-values.Therefore, this is a vertical reflection.Answer: C
8.
Item 4. This is a graph of the function \(g(x)=-3x+2\). Determine the domain value when the range value is \(-4\).
Solution:The range value is the \(y\)-value, so set \(g(x)=-4\).\[-4=-3x+2\]\[-6=-3x\]\[x=2\]Therefore, the domain value is \(2\).Answer: B
9.
Item 6. Which notation best represents the following line graph?
Solution:The filled circle at \(-4\) means \(-4\) is included, and the arrow points left, so \(x\le -4\).The filled circle at \(6\) means \(6\) is included, and the arrow points right, so \(x\ge 6\).Therefore, the notation is \(x\le -4\text{ or }x\ge 6\).Answer: B
10.
Item 12. Write the equation of the graph after it has been expanded vertically by a factor of \(12\) from the original graph of \(y=5^{x-1}\).
Solution:A vertical expansion by a factor of \(12\) means multiply the whole function by \(12\).Start with \(y=5^{x-1}\).\[y=12(5^{x-1})\]Answer: A
11.
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 \(x\):\[3x-12=3(x-4)\]Substitute back:\[y=-2[3(x-4)]^2-6\]Answer: B
12.
Item 9. If \(x-7\) replaces \(x\), and \(y+5\) replaces \(y\), describe the effect on the graph of \(y=f(x)\).
Solution:Replacing \(x\) with \(x-7\) causes a horizontal shift \(7\) units right.Replacing \(y\) with \(y+5\) causes a vertical shift \(5\) units down.Therefore, the graph moves \(7\) units right and \(5\) units down.Answer: C
13.
Item 10. If point \(C(8,-2)\) is on the graph of \(y=f(x)\), which point \(C'\) must be on the transformed graph \(y=\frac{1}{2}f(x)\)?
Solution:The transformation \(y=\frac{1}{2}f(x)\) is a vertical compression by a factor of \(\frac{1}{2}\).The \(x\)-coordinate stays the same, and the \(y\)-coordinate is multiplied by \(\frac{1}{2}\).\[(8,-2)\rightarrow\left(8,-2\cdot\frac{1}{2}\right)=(8,-1)\]Answer: A
14.
Item 15. Which of the following statement is false for a horizontal reflection?
Solution:A horizontal reflection is a reflection across the \(y\)-axis.Points on the \(y\)-axis remain fixed, so invariant points have \(x=0\).These are the \(y\)-intercepts, not the \(x\)-intercepts.Therefore, the false statement is that invariant points are the x-intercepts.Answer: A
15.
Item 3. What is the maximum number of x-intercepts for a cubic function?
Solution:A cubic function has degree \(3\).The maximum number of \(x\)-intercepts of a polynomial is equal to its degree.Therefore, a cubic function can have at most \(3\) \(x\)-intercepts.Answer: B
16.
Item 14. The point \(A(-2,3)\) 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.\[(-2,3)\rightarrow(3,-2)\]Answer: B
17.
Item 19. The equation of \(y=x^2\) is transformed to \(y=-4(x+5)^2+1\). Give the appropriate order of transformations.
Solution:The negative sign in front causes a vertical reflection.The factor \(4\) in front causes a vertical expansion by a factor of \(4\).The \(x+5\) inside causes a horizontal translation \(5\) units left.The \(+1\) outside causes a vertical translation \(1\) unit up.Therefore, the order is vertical reflection, vertical expansion, then horizontal and vertical translations.Answer: A
18.
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
19.
Item 7. Given the graph of \(y=f(x)\), which of the following is the graph of \(y=f(x+2)-1\)?
Solution:Replacing \(x\) with \(x+2\) shifts the graph \(2\) units left.Subtracting \(1\) outside the function shifts the graph \(1\) unit down.The graph showing a shift left \(2\) and down \(1\) is Graph A.Answer: A
20.
Item 20. Identify the order of transformations required to create graph B from graph A.
Solution:From graph A to graph B, first apply a horizontal reflection.Then translate the graph \(5\) units to the right.Therefore, the correct order is horizontal reflection followed by horizontal translation \(5\) units right.Answer: A
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