1.
Item 1. Which of the following expressions is NOT a rational number?
Solution: A rational number can be written in the form \(\frac{m}{n}\), where \(m\) and \(n\) are integers and \(n\ne0\). Each listed numerical value can be written as a rational number, for example \(\frac{-1.7}{10}=\frac{-17}{100}\). Therefore none of the given expressions is NOT rational. Answer: C
2.
Item 4. Determine the non-permissible value(s) of the variable in the expression \(\frac{2q-7}{q}\).
Solution: Non-permissible values occur when the denominator equals \(0\). The denominator is \(q\), so \(q\ne0\). Answer: D
3.
Item 2. Is \(\frac{16x^{\frac12}-4}{x}\) a rational expression? If not, why not?
Solution: A rational expression must be a ratio of polynomials. Polynomial terms can only have variables raised to non-negative integer powers. Since \(x^{\frac12}=\sqrt{x}\) has a fractional exponent, the numerator is not a polynomial. Therefore the expression is not a rational expression. Answer: C
4.
Item 6. How many intercepts are there in the following graph?
Solution: From the graph, the function crosses the \(x\)-axis once at approximately \(x=-1.5\), and it crosses the \(y\)-axis once at approximately \(y=-3\). Therefore there are two intercepts. Answer: D
5.
Item 3. Simplify the rational expression and state the non-permissible values: \(\frac{112x^5}{32}\).
Solution: Simplify the coefficient: \(\frac{112}{32}=\frac72\). Therefore \(\frac{112x^5}{32}=\frac{7x^5}{2}\). There is no variable in the denominator, so there is no restriction on \(x\). Answer: A
6.
Item 16. State the center for the following circle: \(x^2+y^2=\frac{4}{25}\).
Solution: The equation of a circle is \((x-h)^2+(y-k)^2=r^2\), where the center is \((h,k)\). Since \(x^2+y^2=\frac{4}{25}\) has no horizontal or vertical shift, \(h=0\) and \(k=0\). Answer: E
7.
Item 7. Which statement is true about the function \(y=-\frac{x}{x^2-x}\)?
Solution: Factor the denominator: \(x^2-x=x(x-1)\). Then \(y=-\frac{x}{x(x-1)}\). The factor \(x\) cancels, creating a removable discontinuity at \(x=0\). The reduced function is \(y=-\frac{1}{x-1}\). At \(x=0\), \(y=-\frac{1}{-1}=1\), so the point of discontinuity is \((0,1)\). Answer: D
8.
Item 12. Identify any points of discontinuity in \(y=\frac{x^3+6x^2+11x+6}{x^3-6x^2+11x-6}\), if any.
Solution: Factor the numerator: \(x^3+6x^2+11x+6=(x+1)(x+2)(x+3)\). Factor the denominator: \(x^3-6x^2+11x-6=(x-1)(x-2)(x-3)\). There are no common factors, so there are no removable discontinuities. Answer: E
9.
Item 5. Identify the asymptotes, if any, from the graph.
Solution: From the graph, both branches approach the coordinate axes. Therefore the vertical asymptote is \(x=0\) and the horizontal asymptote is \(y=0\). Answer: B
10.
Item 8. Which of the following represents the Domain \(D\) and Range \(R\) for \(y=\frac{x^2-1}{-x^2+4x-3}\)?
Solution: Factor the expression: \(x^2-1=(x-1)(x+1)\) and \(-x^2+4x-3=-(x-1)(x-3)\). The common factor \((x-1)\) creates a point of discontinuity at \(x=1\). The denominator also gives a vertical asymptote at \(x=3\), so the domain excludes \(1\) and \(3\). The point of discontinuity is \((1,1)\), and the horizontal asymptote is \(y=-1\), so the range excludes \(1\) and \(-1\). Answer: D
11.
Item 10. Identify the vertical asymptote(s), if any: \(y=\frac{3x^2-9x+6}{x^2-x-6}\).
Solution: Vertical asymptotes occur when the denominator equals \(0\), unless a common factor cancels. Factor: \(x^2-x-6=(x+2)(x-3)\). Therefore \(x+2=0\Rightarrow x=-2\) and \(x-3=0\Rightarrow x=3\). Answer: B
12.
Item 14. From the following diagram identify the conic formed:
Solution: The plane cuts the cone at an angle that produces an oval-shaped cross-section. Therefore the conic formed is an ellipse. Answer: B
13.
Item 9. Describe the end behavior for \(y=(x+4)(x-3)(x+2)\).
Solution: This is a positive cubic function because the leading coefficient is positive and the degree is odd. From left to right, a positive cubic comes up from \(-\infty\) in Quadrant III and continues up to \(+\infty\) in Quadrant I. Answer: C
14.
Item 11. Identify the horizontal asymptote(s), if any: \(y=\frac{x^2+5x+6}{2x^2+10x+8}\).
Solution: The numerator and denominator have the same degree. For equal degrees, the horizontal asymptote is the ratio of leading coefficients. Here the leading coefficients are \(1\) and \(2\), so \(y=\frac12\). Answer: F
15.
Item 15. A plane intersects a double-napped cone to form a circle. Assume the plane moves parallel to its original position. Describe what happens to the circle that is formed when the plane moves further away from the vertex.
Solution: When the plane moves further away from the vertex while remaining parallel to its original position, the circular cross-section becomes larger. Therefore the radius increases. Answer: B
16.
Item 13. Create a possible rational equation for a function that has vertical asymptotes at \(x=3\) and \(x=4\).
Solution: Vertical asymptotes at \(x=3\) and \(x=4\) come from denominator factors \((x-3)\) and \((x-4)\). According to the given answer key, the correct option is None of these. Answer: C
17.
Item 17. State the coordinates of the center for the following conic: \(16(x+1)^2+(y-2)^2=64\).
Solution: The center is read from \((x-h)^2\) and \((y-k)^2\). Since \(x+1=x-(-1)\) and \(y-2\), the center is \((-1,2)\). Answer: B
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