llmstory
Algebra II Final Exam

Section 1: Multiple Choice (No Calculator Allowed)

Instructions: Choose the best answer for each question. Show your work for partial credit where applicable, but only the final answer will be graded. Estimated time: 20 minutes.

1.

Which of the following is a factor of the polynomial `P(x)=x3−2x2−5x+6P(x) = x^3 - 2x^2 - 5x + 6?

Select one option
2.

Simplify the expression: $\log_2(8) + \log_3(9) - \log_5(25)$.

Select one option
3.

What is the exact value of $\sin\left(\frac{5\pi}{6}\right)$?

Select one option
4.

What is the domain of the function $f(x) = \frac{x+3}{x^2-4}$?

Select one option
5.

The equation \$4x^2 - 9y^2 = 36$ represents which type of conic section?

Select one option
6.

Expand and simplify: $(x-2)(x^2+2x+4)$.

Select one option
7.

Solve for $x$: $\log_4(x) = 3$.

Select one option
8.

What is the principal value of $\arctan(1)$ in radians?

Select one option
9.

Simplify the rational expression: $\frac{x^2-9}{x^2-x-6}$.

Select one option
10.

Which equation represents a circle with center $(0,0)$ and radius \$5$?

Select one option

Section 2: Free Response (No Calculator Allowed)

Instructions: Provide detailed, step-by-step solutions for each problem. Simplify all answers completely. Estimated time: 30 minutes.

11.

A. Find all real and complex roots of the polynomial $P(x) = x^4 - 16$. B. Graph the real roots on a number line.

12.

A. Solve for $x$: $\log_3(x-1) + \log_3(x+1) = \log_3(8)$. B. Expand the expression: $\log\left(\frac{x^2\sqrt{y}}{z^3}\right)$.

13.

A. Prove the identity: $\frac{\sin^2\theta}{1-\cos\theta} = 1+\cos\theta$. B. Find all solutions for $\sin(x) = \frac{\sqrt{3}}{2}$ in the interval $[0, 2\pi)$.

Section 3: Multiple Choice (Calculator Allowed)

Instructions: Choose the best answer for each question. You may use a calculator for computations. Estimated time: 25 minutes.

14.

For the function $g(x) = \frac{2x^2-5x-3}{x^2-9}$, what are the equations of the vertical asymptotes, if any, and the coordinates of any holes?

Select one option
15.

An ellipse has foci at $(\pm 3, 0)$ and vertices at $(\pm 5, 0)$. What is the equation of the ellipse?

Select one option
16.

A Ferris wheel has a radius of \$25$ meters and its center is $30 meters above the ground. If it completes one rotation every `\2minutes,whichequationmodelstheheight‘ minutes, which equation models the height `h‘ofarider(inmeters)abovethegroundasafunctionoftime‘` of a rider (in meters) above the ground as a function of time `t Exam Center

(in minutes), assuming the rider starts at the lowest point?

Select one option
17.

The population of a city is $P(t) = 50000e^{0.02t}$, where $t$ is the number of years since 2000. In what year will the population reach \$75000$?

Select one option
18.

Which of the following describes the end behavior of the polynomial $f(x) = -2x^5 + 3x^3 - x + 7$?

Select one option
19.

What is the horizontal asymptote of the function $h(x) = \frac{3x^2+2x-1}{x^2-4x+5}$?

Select one option
20.

The equation $x^2 + 6x + y^2 - 4y = 12$ represents a circle. What are its center and radius?

Select one option
21.

In $\triangle ABC$, $a=10$, $b=15$, and $C=60^\circ$. What is the length of side $c$ to the nearest tenth?

Select one option
22.

If $(x+2)$ is a factor of $f(x) = x^3 + kx^2 - x - 6$, what is the value of $k$?

Select one option
23.

Given $\log_b 2 = 0.35$ and $\log_b 3 = 0.56$, what is $\log_b 12$?

Select one option

Section 4: Free Response (Calculator Allowed)

Instructions: Provide detailed, step-by-step solutions for each problem. You may use a calculator. Clearly show all steps and label any graphs. Estimated time: 45 minutes.

24.

Consider the function $f(x) = \frac{x^2+x-6}{x^2-x-2}$. A. Find the domain of the function. B. Identify any vertical asymptotes and holes. Justify your answer. C. Identify any horizontal or slant asymptotes. Justify your answer. D. Find the x- and y-intercepts. E. Sketch a graph of the function, labeling all intercepts and asymptotes.

25.

Consider the equation \$4x^2 + 9y^2 - 24x + 36y + 36 = 0$. A. Classify the conic section. B. Rewrite the equation in standard form. C. Find the center, vertices, and foci. D. Sketch the graph of the conic section, labeling the center and vertices.

26.

A. Solve the equation \$2\cos^2(x) + \cos(x) - 1 = 0$ for $x$ in the interval $[0, 2\pi)$. B. A buoy bobs up and down in the water. The distance from its highest point to its lowest point is \$40$ cm. It takes $4secondstocompleteonefullcycle.Ifthebuoyisatitsequilibriumpositionandmovingupwardsat‘ seconds to complete one full cycle. If the buoy is at its equilibrium position and moving upwards at `t=0‘,writeanequationforthedisplacement‘`, write an equation for the displacement `d‘(incm)fromitsequilibriumpositionasafunctionoftime‘` (in cm) from its equilibrium position as a function of time `t Exam Center

(in seconds).

Copyright © 2026 llmstory.comPrivacy PolicyTerms of Service
.\",\"type\":\"single-choice\",\"options\":[\"1\",\"2\",\"3\",\"4\"],\"correctAnswer\":\"3\"}]},{\"id\":\"group-3\",\"questions\":[{\"id\":3,\"text\":\"What is the exact value of `$\\\\sin\\\\left(\\\\frac{5\\\\pi}{6}\\\\right) Exam Center
?\",\"type\":\"single-choice\",\"options\":[\"`$-\\\\frac{\\\\sqrt{3}}{2} Exam Center
\",\"`$-\\\\frac{1}{2} Exam Center
\",\"`$\\\\frac{1}{2} Exam Center
\",\"`$\\\\frac{\\\\sqrt{3}}{2} Exam Center
\"],\"correctAnswer\":\"`$\\\\frac{1}{2} Exam Center
\"}]},{\"id\":\"group-4\",\"questions\":[{\"id\":4,\"text\":\"What is the domain of the function `$f(x) = \\\\frac{x+3}{x^2-4} Exam Center
?\",\"type\":\"single-choice\",\"options\":[\"`$\\\\{x | x \\\\neq 2\\\\} Exam Center
\",\"`$\\\\{x | x \\\\neq -2\\\\} Exam Center
\",\"`$\\\\{x | x \\\\neq 2 \\\\text{ and } x \\\\neq -2\\\\} Exam Center
\",\"`All real numbers`\"],\"correctAnswer\":\"`$\\\\{x | x \\\\neq 2 \\\\text{ and } x \\\\neq -2\\\\} Exam Center
\"}]},{\"id\":\"group-5\",\"questions\":[{\"id\":5,\"text\":\"The equation `$4x^2 - 9y^2 = 36 Exam Center
represents which type of conic section?\",\"type\":\"single-choice\",\"options\":[\"Circle\",\"Ellipse\",\"Parabola\",\"Hyperbola\"],\"correctAnswer\":\"Hyperbola\"}]},{\"id\":\"group-6\",\"questions\":[{\"id\":6,\"text\":\"Expand and simplify: `$(x-2)(x^2+2x+4) Exam Center
.\",\"type\":\"single-choice\",\"options\":[\"`$x^3+8 Exam Center
\",\"`$x^3-8 Exam Center
\",\"`$x^3-4x+8 Exam Center
\",\"`$x^3+4x-8 Exam Center
\"],\"correctAnswer\":\"`$x^3-8 Exam Center
\"}]},{\"id\":\"group-7\",\"questions\":[{\"id\":7,\"text\":\"Solve for `$x Exam Center
: `$\\\\log_4(x) = 3 Exam Center
.\",\"type\":\"single-choice\",\"options\":[\"12\",\"64\",\"27\",\"81\"],\"correctAnswer\":\"64\"}]},{\"id\":\"group-8\",\"questions\":[{\"id\":8,\"text\":\"What is the principal value of `$\\\\arctan(1) Exam Center
in radians?\",\"type\":\"single-choice\",\"options\":[\"`$0 Exam Center
\",\"`$\\\\frac{\\\\pi}{4} Exam Center
\",\"`$\\\\frac{\\\\pi}{2} Exam Center
\",\"`$\\\\pi Exam Center
\"],\"correctAnswer\":\"`$\\\\frac{\\\\pi}{4} Exam Center
\"}]},{\"id\":\"group-9\",\"questions\":[{\"id\":9,\"text\":\"Simplify the rational expression: `$\\\\frac{x^2-9}{x^2-x-6} Exam Center
.\",\"type\":\"single-choice\",\"options\":[\"`$\\\\frac{x-3}{x-2} Exam Center
\",\"`$\\\\frac{x+3}{x+2} Exam Center
\",\"`$\\\\frac{x-3}{x+2} Exam Center
\",\"`$\\\\frac{x+3}{x-2} Exam Center
\"],\"correctAnswer\":\"`$\\\\frac{x+3}{x+2} Exam Center
\"}]},{\"id\":\"group-10\",\"questions\":[{\"id\":10,\"text\":\"Which equation represents a circle with center `$(0,0) Exam Center
and radius `$5 Exam Center
?\",\"type\":\"single-choice\",\"options\":[\"`$x^2+y^2=5 Exam Center
\",\"`$x^2-y^2=25 Exam Center
\",\"`$x^2+y^2=25 Exam Center
\",\"`$x^2+y^2=10 Exam Center
\"],\"correctAnswer\":\"`$x^2+y^2=25 Exam Center
\"}]},{\"id\":\"group-11\",\"passage\":\"Section 2: Free Response (No Calculator Allowed)\\n\\nInstructions: Provide detailed, step-by-step solutions for each problem. Simplify all answers completely. Estimated time: 30 minutes.\",\"questions\":[{\"id\":11,\"text\":\"A. Find all real and complex roots of the polynomial `$P(x) = x^4 - 16 Exam Center
.\\nB. Graph the real roots on a number line.\",\"type\":\"short-answer\",\"correctAnswer\":\"A. To find the roots of `$P(x) = x^4 - 16 Exam Center
, set `$P(x)=0 Exam Center
: \\n`$x^4 - 16 = 0 Exam Center
\\n`$(x^2 - 4)(x^2 + 4) = 0 Exam Center
\\n`$(x-2)(x+2)(x^2+4) = 0 Exam Center
\\nSetting each factor to zero:\\n`$x-2=0 \\\\implies x=2 Exam Center
\\n`$x+2=0 \\\\implies x=-2 Exam Center
\\n`$x^2+4=0 \\\\implies x^2=-4 \\\\implies x = \\\\pm\\\\sqrt{-4} \\\\implies x = \\\\pm 2i Exam Center
\\nThe roots are `$2, -2, 2i, -2i Exam Center
.\\n\\nB. Real roots on a number line:\\n\\n```\\n \u003c-----|-----|-----\u003e\\n -2 0 2\\n```\"}]},{\"id\":\"group-12\",\"questions\":[{\"id\":12,\"text\":\"A. Solve for `$x Exam Center
: `$\\\\log_3(x-1) + \\\\log_3(x+1) = \\\\log_3(8) Exam Center
.\\nB. Expand the expression: `$\\\\log\\\\left(\\\\frac{x^2\\\\sqrt{y}}{z^3}\\\\right) Exam Center
.\",\"type\":\"short-answer\",\"correctAnswer\":\"A. `$\\\\log_3(x-1) + \\\\log_3(x+1) = \\\\log_3(8) Exam Center
\\n`$\\\\log_3((x-1)(x+1)) = \\\\log_3(8) Exam Center
\\n`$\\\\log_3(x^2-1) = \\\\log_3(8) Exam Center
\\n`$x^2-1 = 8 Exam Center
\\n`$x^2 = 9 Exam Center
\\n`$x = \\\\pm 3 Exam Center
\\nCheck for extraneous solutions: For `$\\\\log_3(x-1) Exam Center
to be defined, `$x-1 \u003e 0 \\\\implies x \u003e 1 Exam Center
. For `$\\\\log_3(x+1) Exam Center
to be defined, `$x+1 \u003e 0 \\\\implies x \u003e -1 Exam Center
. Both require `$x \u003e 1 Exam Center
. Therefore, `$x=3 Exam Center
is the only valid solution. `$x=-3 Exam Center
is extraneous.\\nSolution: `$x=3 Exam Center
.\\n\\nB. `$\\\\log\\\\left(\\\\frac{x^2\\\\sqrt{y}}{z^3}\\\\right) = \\\\log(x^2\\\\sqrt{y}) - \\\\log(z^3) Exam Center
\\n`$= \\\\log(x^2) + \\\\log(\\\\sqrt{y}) - \\\\log(z^3) Exam Center
\\n`$= 2\\\\log(x) + \\\\frac{1}{2}\\\\log(y) - 3\\\\log(z) Exam Center
\"}]},{\"id\":\"group-13\",\"questions\":[{\"id\":13,\"text\":\"A. Prove the identity: `$\\\\frac{\\\\sin^2\\\\theta}{1-\\\\cos\\\\theta} = 1+\\\\cos\\\\theta Exam Center
.\\nB. Find all solutions for `$\\\\sin(x) = \\\\frac{\\\\sqrt{3}}{2} Exam Center
in the interval `$[0, 2\\\\pi) Exam Center
.\",\"type\":\"short-answer\",\"correctAnswer\":\"A. Start with the Left Hand Side (LHS):\\n`LHS` `$= \\\\frac{\\\\sin^2\\\\theta}{1-\\\\cos\\\\theta} Exam Center
\\nUsing the Pythagorean identity `$\\\\sin^2\\\\theta = 1-\\\\cos^2\\\\theta Exam Center
:\\n`$= \\\\frac{1-\\\\cos^2\\\\theta}{1-\\\\cos\\\\theta} Exam Center
\\nFactor the numerator (difference of squares):\\n`$= \\\\frac{(1-\\\\cos\\\\theta)(1+\\\\cos\\\\theta)}{1-\\\\cos\\\\theta} Exam Center
\\nCancel the common factor `$(1-\\\\cos\\\\theta) Exam Center
(assuming `$\\\\cos\\\\theta \\\\neq 1 Exam Center
):\\n`$= 1+\\\\cos\\\\theta Exam Center
\\n`$= RHS Exam Center
\\nThus, the identity is proven.\\n\\nB. For `$\\\\sin(x) = \\\\frac{\\\\sqrt{3}}{2} Exam Center
in the interval `$[0, 2\\\\pi) Exam Center
:\\nThe reference angle is `$\\\\frac{\\\\pi}{3} Exam Center
because `$\\\\sin\\\\left(\\\\frac{\\\\pi}{3}\\\\right) = \\\\frac{\\\\sqrt{3}}{2} Exam Center
.\\nSine is positive in Quadrants I and II.\\nQuadrant I solution: `$x = \\\\frac{\\\\pi}{3} Exam Center
.\\nQuadrant II solution: `$x = \\\\pi - \\\\frac{\\\\pi}{3} = \\\\frac{3\\\\pi}{3} - \\\\frac{\\\\pi}{3} = \\\\frac{2\\\\pi}{3} Exam Center
.\\nThe solutions are `$\\\\frac{\\\\pi}{3} Exam Center
and `$\\\\frac{2\\\\pi}{3} Exam Center
.\"}]},{\"id\":\"group-14\",\"passage\":\"Section 3: Multiple Choice (Calculator Allowed)\\n\\nInstructions: Choose the best answer for each question. You may use a calculator for computations. Estimated time: 25 minutes.\",\"questions\":[{\"id\":14,\"text\":\"For the function `$g(x) = \\\\frac{2x^2-5x-3}{x^2-9} Exam Center
, what are the equations of the vertical asymptotes, if any, and the coordinates of any holes?\",\"type\":\"single-choice\",\"options\":[\"VA: `$x=3 Exam Center
, Hole: none\",\"VA: `$x=-3 Exam Center
, Hole: `$(3, \\\\frac{7}{6}) Exam Center
\",\"VA: `$x=3, x=-3 Exam Center
, Hole: none\",\"VA: `$x=-3 Exam Center
, Hole: none\"],\"correctAnswer\":\"VA: `$x=-3 Exam Center
, Hole: `$(3, \\\\frac{7}{6}) Exam Center
\"}]},{\"id\":\"group-15\",\"questions\":[{\"id\":15,\"text\":\"An ellipse has foci at `$(\\\\pm 3, 0) Exam Center
and vertices at `$(\\\\pm 5, 0) Exam Center
. What is the equation of the ellipse?\",\"type\":\"single-choice\",\"options\":[\"`$\\\\frac{x^2}{25} + \\\\frac{y^2}{16} = 1 Exam Center
\",\"`$\\\\frac{x^2}{16} + \\\\frac{y^2}{25} = 1 Exam Center
\",\"`$\\\\frac{x^2}{25} - \\\\frac{y^2}{16} = 1 Exam Center
\",\"`$\\\\frac{x^2}{16} - \\\\frac{y^2}{25} = 1 Exam Center
\"],\"correctAnswer\":\"`$\\\\frac{x^2}{25} + \\\\frac{y^2}{16} = 1 Exam Center
\"}]},{\"id\":\"group-16\",\"questions\":[{\"id\":16,\"text\":\"A Ferris wheel has a radius of `$25$ meters and its center is `$30$ meters above the ground. If it completes one rotation every `$2$ minutes, which equation models the height `$h Exam Center
of a rider (in meters) above the ground as a function of time `$t Exam Center
(in minutes), assuming the rider starts at the lowest point?\",\"type\":\"single-choice\",\"options\":[\"`$h(t) = -25\\\\cos(\\\\pi t) + 30 Exam Center
\",\"`$h(t) = 25\\\\sin(\\\\pi t) + 30 Exam Center
\",\"`$h(t) = -25\\\\cos(2\\\\pi t) + 30 Exam Center
\",\"`$h(t) = 25\\\\cos(\\\\pi t) + 30 Exam Center
\"],\"correctAnswer\":\"`$h(t) = -25\\\\cos(\\\\pi t) + 30 Exam Center
\"}]},{\"id\":\"group-17\",\"questions\":[{\"id\":17,\"text\":\"The population of a city is `$P(t) = 50000e^{0.02t} Exam Center
, where `$t Exam Center
is the number of years since 2000. In what year will the population reach `$75000 Exam Center
?\",\"type\":\"single-choice\",\"options\":[\"2010\",\"2015\",\"2020\",\"2025\"],\"correctAnswer\":\"2020\"}]},{\"id\":\"group-18\",\"questions\":[{\"id\":18,\"text\":\"Which of the following describes the end behavior of the polynomial `$f(x) = -2x^5 + 3x^3 - x + 7 Exam Center
?\",\"type\":\"single-choice\",\"options\":[\"As `$x \\\\to \\\\infty, f(x) \\\\to \\\\infty Exam Center
and as `$x \\\\to -\\\\infty, f(x) \\\\to \\\\infty Exam Center
\",\"As `$x \\\\to \\\\infty, f(x) \\\\to -\\\\infty Exam Center
and as `$x \\\\to -\\\\infty, f(x) \\\\to -\\\\infty Exam Center
\",\"As `$x \\\\to \\\\infty, f(x) \\\\to \\\\infty Exam Center
and as `$x \\\\to -\\\\infty, f(x) \\\\to -\\\\infty Exam Center
\",\"As `$x \\\\to \\\\infty, f(x) \\\\to -\\\\infty Exam Center
and as `$x \\\\to -\\\\infty, f(x) \\\\to \\\\infty Exam Center
\"],\"correctAnswer\":\"As `$x \\\\to \\\\infty, f(x) \\\\to -\\\\infty Exam Center
and as `$x \\\\to -\\\\infty, f(x) \\\\to \\\\infty Exam Center
\"}]},{\"id\":\"group-19\",\"questions\":[{\"id\":19,\"text\":\"What is the horizontal asymptote of the function `$h(x) = \\\\frac{3x^2+2x-1}{x^2-4x+5} Exam Center
?\",\"type\":\"single-choice\",\"options\":[\"`$y=0 Exam Center
\",\"`$y=1 Exam Center
\",\"`$y=3 Exam Center
\",\"No horizontal asymptote\"],\"correctAnswer\":\"`$y=3 Exam Center
\"}]},{\"id\":\"group-20\",\"questions\":[{\"id\":20,\"text\":\"The equation `$x^2 + 6x + y^2 - 4y = 12 Exam Center
represents a circle. What are its center and radius?\",\"type\":\"single-choice\",\"options\":[\"Center: `$(3, -2) Exam Center
, Radius: `$5 Exam Center
\",\"Center: `$(3, -2) Exam Center
, Radius: `$25 Exam Center
\",\"Center: `$(-3, 2) Exam Center
, Radius: `$5 Exam Center
\",\"Center: `$(-3, 2) Exam Center
, Radius: `$25 Exam Center
\"],\"correctAnswer\":\"Center: `$(-3, 2) Exam Center
, Radius: `$5 Exam Center
\"}]},{\"id\":\"group-21\",\"questions\":[{\"id\":21,\"text\":\"In `$\\\\triangle ABC Exam Center
, `$a=10 Exam Center
, `$b=15 Exam Center
, and `$C=60^\\\\circ Exam Center
. What is the length of side `$c Exam Center
to the nearest tenth?\",\"type\":\"single-choice\",\"options\":[\"7.5\",\"8.7\",\"12.8\",\"13.2\"],\"correctAnswer\":\"13.2\"}]},{\"id\":\"group-22\",\"questions\":[{\"id\":22,\"text\":\"If `$(x+2) Exam Center
is a factor of `$f(x) = x^3 + kx^2 - x - 6 Exam Center
, what is the value of `$k Exam Center
?\",\"type\":\"single-choice\",\"options\":[\"-3\",\"-2\",\"2\",\"3\"],\"correctAnswer\":\"3\"}]},{\"id\":\"group-23\",\"questions\":[{\"id\":23,\"text\":\"Given `$\\\\log_b 2 = 0.35 Exam Center
and `$\\\\log_b 3 = 0.56 Exam Center
, what is `$\\\\log_b 12 Exam Center
?\",\"type\":\"single-choice\",\"options\":[\"0.70\",\"0.91\",\"1.26\",\"1.47\"],\"correctAnswer\":\"1.26\"}]},{\"id\":\"group-24\",\"passage\":\"Section 4: Free Response (Calculator Allowed)\\n\\nInstructions: Provide detailed, step-by-step solutions for each problem. You may use a calculator. Clearly show all steps and label any graphs. Estimated time: 45 minutes.\",\"questions\":[{\"id\":24,\"text\":\"Consider the function `$f(x) = \\\\frac{x^2+x-6}{x^2-x-2} Exam Center
.\\nA. Find the domain of the function.\\nB. Identify any vertical asymptotes and holes. Justify your answer.\\nC. Identify any horizontal or slant asymptotes. Justify your answer.\\nD. Find the x- and y-intercepts.\\nE. Sketch a graph of the function, labeling all intercepts and asymptotes.\",\"type\":\"short-answer\",\"correctAnswer\":\"A. To find the domain, set the denominator to zero:\\n`$x^2-x-2 = 0 Exam Center
\\n`$(x-2)(x+1) = 0 Exam Center
\\n`$x=2 Exam Center
or `$x=-1 Exam Center
\\nDomain: `$\\\\{x | x \\\\neq 2 \\\\text{ and } x \\\\neq -1\\\\} Exam Center
.\\n\\nB. Factor the numerator: `$x^2+x-6 = (x+3)(x-2) Exam Center
.\\nSo, `$f(x) = \\\\frac{(x+3)(x-2)}{(x-2)(x+1)} Exam Center
.\\nThere is a common factor of `$(x-2) Exam Center
, so there is a hole at `$x=2 Exam Center
. To find the y-coordinate of the hole, simplify the function to `$g(x) = \\\\frac{x+3}{x+1} Exam Center
for `$x \\\\neq 2 Exam Center
and substitute `$x=2 Exam Center
:\\n`$g(2) = \\\\frac{2+3}{2+1} = \\\\frac{5}{3} Exam Center
.\\nHole at `$(2, \\\\frac{5}{3}) Exam Center
.\\n\\nFor vertical asymptotes, set the simplified denominator to zero: `$x+1=0 \\\\implies x=-1 Exam Center
. \\nVertical Asymptote: `$x=-1 Exam Center
.\\n\\nC. Since the degree of the numerator (2) is equal to the degree of the denominator (2), there is a horizontal asymptote. It is the ratio of the leading coefficients:\\nHorizontal Asymptote: `$y=\\\\frac{1}{1} = 1 Exam Center
.\\nThere is no slant asymptote because the degree of the numerator is not exactly one greater than the degree of the denominator.\\n\\nD. Y-intercept: Set `$x=0 Exam Center
in the original function (or simplified if no hole at x=0):\\n`$f(0) = \\\\frac{0^2+0-6}{0^2-0-2} = \\\\frac{-6}{-2} = 3 Exam Center
.\\nY-intercept: `$(0,3) Exam Center
.\\n\\nX-intercepts: Set the numerator of the simplified function to zero (after canceling common factors):\\n`$x+3=0 \\\\implies x=-3 Exam Center
.\\nX-intercept: `$(-3,0) Exam Center
.\\n\\nE. Sketch of the graph:\\n* Plot intercepts: `(-3,0)` and `(0,3)`.\\n* Draw vertical asymptote `$x=-1 Exam Center
.\\n* Draw horizontal asymptote `$y=1 Exam Center
.\\n* Plot the hole at `$(2, \\\\frac{5}{3}) Exam Center
(approximately `$(2, 1.67) Exam Center
).\\n* The graph approaches the asymptotes. For `$x \u003c -1 Exam Center
, `$(x+3) Exam Center
is positive, `$(x+1) Exam Center
is negative, so `$f(x) Exam Center
is negative. As `$x \\\\to -1^- Exam Center
, `$f(x) \\\\to -\\\\infty Exam Center
. As `$x \\\\to -\\\\infty Exam Center
, `$f(x) \\\\to 1^+ Exam Center
.\\n* For `$x \u003e -1 Exam Center
, `$(x+3) Exam Center
is positive, `$(x+1) Exam Center
is positive, so `$f(x) Exam Center
is positive. As `$x \\\\to -1^+ Exam Center
, `$f(x) \\\\to \\\\infty Exam Center
. As `$x \\\\to \\\\infty Exam Center
, `$f(x) \\\\to 1^- Exam Center
. \\n* The graph passes through `(-3,0)` and `(0,3)`. The hole is above the HA.\"}]},{\"id\":\"group-25\",\"questions\":[{\"id\":25,\"text\":\"Consider the equation `$4x^2 + 9y^2 - 24x + 36y + 36 = 0 Exam Center
.\\nA. Classify the conic section.\\nB. Rewrite the equation in standard form.\\nC. Find the center, vertices, and foci.\\nD. Sketch the graph of the conic section, labeling the center and vertices.\",\"type\":\"short-answer\",\"correctAnswer\":\"A. The conic section is an **Ellipse** because the `$x^2 Exam Center
and `$y^2 Exam Center
terms both have positive coefficients and are different. (Same positive coefficients would be a circle).\\n\\nB. To rewrite in standard form, complete the square:\\n`$4x^2 - 24x + 9y^2 + 36y = -36 Exam Center
\\n`$4(x^2 - 6x) + 9(y^2 + 4y) = -36 Exam Center
\\n`$4(x^2 - 6x + 9) + 9(y^2 + 4y + 4) = -36 + 4(9) + 9(4) Exam Center
\\n`$4(x-3)^2 + 9(y+2)^2 = -36 + 36 + 36 Exam Center
\\n`$4(x-3)^2 + 9(y+2)^2 = 36 Exam Center
\\nDivide by 36:\\n`$\\\\frac{4(x-3)^2}{36} + \\\\frac{9(y+2)^2}{36} = \\\\frac{36}{36} Exam Center
\\n`$\\\\frac{(x-3)^2}{9} + \\\\frac{(y+2)^2}{4} = 1 Exam Center
\\n\\nC. From the standard form `$\\\\frac{(x-h)^2}{a^2} + \\\\frac{(y-k)^2}{b^2} = 1 Exam Center
:\\nCenter: `$(h,k) = (3, -2) Exam Center
.\\n\\nSince `$a^2=9 Exam Center
and `$b^2=4 Exam Center
, we have `$a=3 Exam Center
and `$b=2 Exam Center
. The major axis is horizontal because `$a^2 Exam Center
is under the `$x Exam Center
term.\\nVertices: `$(h \\\\pm a, k) = (3 \\\\pm 3, -2) Exam Center
\\nVertices: `$(0, -2) Exam Center
and `$(6, -2) Exam Center
.\\n\\nTo find the foci, use `$c^2 = a^2 - b^2 Exam Center
:\\n`$c^2 = 9 - 4 = 5 Exam Center
\\n`$c = \\\\sqrt{5} Exam Center
.\\nFoci: `$(h \\\\pm c, k) = (3 \\\\pm \\\\sqrt{5}, -2) Exam Center
\\nFoci: `$(3 - \\\\sqrt{5}, -2) Exam Center
and `$(3 + \\\\sqrt{5}, -2) Exam Center
(approximately `$(3 - 2.236, -2) = (0.764, -2) Exam Center
and `$(3 + 2.236, -2) = (5.236, -2) Exam Center
).\\n\\nD. Sketch of the graph:\\n* Plot the center `$(3, -2) Exam Center
.\\n* Plot the vertices `$(0, -2) Exam Center
and `$(6, -2) Exam Center
.\\n* Since `$b=2 Exam Center
, plot co-vertices at `$(3, -2 \\\\pm 2) Exam Center
, which are `$(3, 0) Exam Center
and `$(3, -4) Exam Center
.\\n* Sketch the ellipse passing through the vertices and co-vertices.\\n* Label the center and vertices.\"}]},{\"id\":\"group-26\",\"questions\":[{\"id\":26,\"text\":\"A. Solve the equation `$2\\\\cos^2(x) + \\\\cos(x) - 1 = 0 Exam Center
for `$x Exam Center
in the interval `$[0, 2\\\\pi) Exam Center
.\\nB. A buoy bobs up and down in the water. The distance from its highest point to its lowest point is `$40$ cm. It takes `$4$ seconds to complete one full cycle. If the buoy is at its equilibrium position and moving upwards at `$t=0 Exam Center
, write an equation for the displacement `$d Exam Center
(in cm) from its equilibrium position as a function of time `$t Exam Center
(in seconds).\",\"type\":\"short-answer\",\"correctAnswer\":\"A. Let `$u = \\\\cos(x) Exam Center
. The equation becomes `$2u^2 + u - 1 = 0 Exam Center
.\\nFactor the quadratic equation:\\n`$(2u-1)(u+1) = 0 Exam Center
\\nSet each factor to zero:\\n`$2u-1 = 0 \\\\implies u = \\\\frac{1}{2} Exam Center
\\n`$u+1 = 0 \\\\implies u = -1 Exam Center
\\n\\nSubstitute back `$\\\\cos(x) Exam Center
for `$u Exam Center
:\\nCase 1: `$\\\\cos(x) = \\\\frac{1}{2} Exam Center
\\nIn the interval `$[0, 2\\\\pi) Exam Center
, cosine is positive in Quadrants I and IV.\\nQuadrant I solution: `$x = \\\\frac{\\\\pi}{3} Exam Center
.\\nQuadrant IV solution: `$x = 2\\\\pi - \\\\frac{\\\\pi}{3} = \\\\frac{5\\\\pi}{3} Exam Center
.\\n\\nCase 2: `$\\\\cos(x) = -1 Exam Center
\\nIn the interval `$[0, 2\\\\pi) Exam Center
, `$\\\\cos(x) = -1 Exam Center
when `$x = \\\\pi Exam Center
.\\n\\nCombining all solutions, the solutions for `$x Exam Center
in the interval `$[0, 2\\\\pi) Exam Center
are `$\\\\frac{\\\\pi}{3}, \\\\pi, \\\\frac{5\\\\pi}{3} Exam Center
.\\n\\nB. The distance from highest to lowest point is `$40$ cm, so the amplitude `$A = \\\\frac{40}{2} = 20$ cm.\\nThe period is `$4$ seconds. The angular frequency `$\\\\omega = \\\\frac{2\\\\pi}{\\\\text{Period}} = \\\\frac{2\\\\pi}{4} = \\\\frac{\\\\pi}{2} Exam Center
.\\nSince the buoy is at its equilibrium position and moving upwards at `$t=0 Exam Center
, a sine function is appropriate (as sine starts at equilibrium and increases).\\nThe general form is `$d(t) = A\\\\sin(\\\\omega t) Exam Center
.\\nSubstituting the values:\\n`$d(t) = 20\\\\sin\\\\left(\\\\frac{\\\\pi}{2}t\\\\right) Exam Center
.\"}]}],\"targetAudience\":\"11th Grade Algebra II Students\",\"creationTime\":1789326064930,\"lastVisited\":1789326064930,\"model\":\"predefined\",\"timeTaken\":0,\"apiProvider\":\"static\"}},\"actionData\":null,\"errors\":null}");