This exam was created by a large language model. Please use it with caution, as inaccuracies may be present. Always verify critical information.
1.
Which of the following expressions is equivalent to $(x^3 - 2x^2 - 5x + 6) \div (x - 3)$?
2.
Simplify the expression: $\log_2(8x) - \log_2(x)$.
3.
What are the vertical asymptotes of the function $f(x) = \frac{x - 1}{(x + 2)(x - 3)}$?
4.
The equation $4x^2 - 9y^2 = 36$ represents which type of conic section?
5.
What is the exact value of $\cos(\frac{5\pi}{6})$?
6.
Describe the end behavior of the polynomial function $P(x) = -2x^4 + 3x^3 - x + 5$.
7.
Which of the following is the inverse of the function $f(x) = 3^x$?
8.
For the rational function $g(x) = \frac{x + 4}{x^2 - 16}$, what is the domain?
9.
What is the center and radius of the circle given by the equation $x^2 + y^2 - 6x + 8y - 11 = 0$?
10.
Which of the following is equivalent to $\tan(\theta)$?
11.
Consider the polynomial function $P(x) = x^3 - 4x^2 - 11x + 30$.
Graph this function by finding its x-intercepts (roots), y-intercept, and describing its end behavior. Sketch the graph, clearly labeling the intercepts.
12.
Solve the logarithmic equation for x: $\log_5(x) + \log_5(x - 4) = 1$.
Show all steps and check for extraneous solutions.
13.
Subtract the following rational expressions and simplify your answer: $\frac{x}{x^2 - 4} - \frac{2}{x + 2}$.
Show all steps.
14.
Graph the parabola given by the equation $y = x^2 - 4x + 3$.
Clearly label the vertex and any intercepts.
15.
Solve the trigonometric equation $2\sin(\theta) - 1 = 0$ for $0 \le \theta < 2\pi$.
Show all steps.
16.
When the polynomial $P(x) = 3x^3 - 2x^2 + x - 5$ is divided by $(x + 1)$, what is the remainder?
17.
Using a calculator, evaluate $\log_7(50)$ to four decimal places.
18.
Solve the equation $\frac{3}{x} + \frac{1}{2} = \frac{x + 1}{x}$.
19.
A parabolic satellite dish has a cross-section modeled by the equation $y = \frac{1}{16}x^2$. If the receiver is placed at the focus of the parabola, how far is the receiver from the vertex?
20.
A ladder is leaning against a wall. The ladder is 15 feet long and forms an angle of 65∘ with the ground. How high up the wall does the ladder reach? Round to the nearest tenth of a foot.
21.
Which of the following is a root of the polynomial $P(x) = x^4 - 6x^3 + 12x^2 - 10x + 3$?
22.
The population of a certain bacteria doubles every 3 hours. If you start with 100 bacteria, how long will it take for the population to reach 10,000 bacteria? Use the formula $N(t) = N_0 \cdot 2^{t/d}$ where d is the doubling time. Round to the nearest hour.
23.
What are the equations of the vertical and horizontal asymptotes for the function $f(x) = \frac{2x^2 - 5x + 3}{x^2 - 4}$?
24.
The equation $x^2 + y^2 - 4x + 6y - 3 = 0$ represents a circle. What are its center and radius?
25.
What is the amplitude and period of the function $f(x) = -3\cos(2x + \pi) + 1$?
26.
Analyze the polynomial function $f(x) = x^4 - x^3 - 7x^2 + x + 6$ by finding all its real roots, its y-intercept, and describing its end behavior. Then, sketch a detailed graph of the function, clearly labeling all intercepts.
27.
Solve the equation $e^{2x} - 5e^x + 6 = 0$ for x. Round your answers to three decimal places where appropriate.
Show all steps.
28.
Analyze the rational function $f(x) = \frac{x^2 - 9}{x^2 - x - 6}$.
Identify all holes, vertical asymptotes, horizontal asymptotes, x-intercepts, and y-intercepts. Then, sketch a detailed graph of the function, clearly labeling all these features.
29.
Consider the equation $4x^2 + 9y^2 = 36$.
Identify the type of conic section, convert it to standard form, find its center, vertices, and foci. Then, sketch the graph, clearly labeling the center, vertices, and foci.
30.
Graph the trigonometric function $y = 2\sin(2x - \pi) + 1$.
Identify the amplitude, period, phase shift, and vertical shift. Show at least one full period of the graph, clearly labeling key points (e.g., maximums, minimums, and intercepts or points on the midline).
?\",\"type\":\"single-choice\",\"options\":[\"$x^2 + x - 2$\",\"$x^2 - 5x + 11$\",\"$x^2 + x - 2 + \\\\frac{0}{x-3}$\",\"$x^2 - 5x + 11 - \\\\frac{27}{x-3}$\"],\"correctAnswer\":\"C\"}]},{\"id\":\"group-2\",\"questions\":[{\"id\":2,\"text\":\"Simplify the expression: `$\\\\log_2(8x) - \\\\log_2(x)
Exam Center
.\",\"type\":\"single-choice\",\"options\":[\"$3$\",\"$\\\\log_2(7x)$\",\"$\\\\log_2(8)$\",\"$\\\\log_2(8 - x)$\"],\"correctAnswer\":\"A\"}]},{\"id\":\"group-3\",\"questions\":[{\"id\":3,\"text\":\"What are the vertical asymptotes of the function `$f(x) = \\\\frac{x - 1}{(x + 2)(x - 3)}$`?\",\"type\":\"single-choice\",\"options\":[\"$x = 1$\",\"$x = -2$\",\"$x = 3$\",\"$x = -2$ and $x = 3$\"],\"correctAnswer\":\"D\"}]},{\"id\":\"group-4\",\"questions\":[{\"id\":4,\"text\":\"The equation `$4x^2 - 9y^2 = 36$` represents which type of conic section?\",\"type\":\"single-choice\",\"options\":[\"Circle\",\"Hyperbola\",\"Parabola\",\"Ellipse\"],\"correctAnswer\":\"B\"}]},{\"id\":\"group-5\",\"questions\":[{\"id\":5,\"text\":\"What is the exact value of `$\\\\cos(\\\\frac{5\\\\pi}{6})
Exam Center
?\",\"type\":\"single-choice\",\"options\":[\"$\\\\frac{1}{2}$\",\"$\\\\frac{\\\\sqrt{3}}{2}$\",\"$-\\\\frac{\\\\sqrt{3}}{2}$\",\"$-\\\\frac{1}{2}$\"],\"correctAnswer\":\"C\"}]},{\"id\":\"group-6\",\"questions\":[{\"id\":6,\"text\":\"Describe the end behavior of the polynomial function `$P(x) = -2x^4 + 3x^3 - x + 5$`.\",\"type\":\"single-choice\",\"options\":[\"As $x \\\\to \\\\infty$, $P(x) \\\\to -\\\\infty$ and as $x \\\\to -\\\\infty$, $P(x) \\\\to -\\\\infty$.\",\"As $x \\\\to \\\\infty$, $P(x) \\\\to \\\\infty$ and as $x \\\\to -\\\\infty$, $P(x) \\\\to \\\\infty$.\",\"As $x \\\\to \\\\infty$, $P(x) \\\\to -\\\\infty$ and as $x \\\\to -\\\\infty$, $P(x) \\\\to \\\\infty$.\",\"As $x \\\\to \\\\infty$, $P(x) \\\\to \\\\infty$ and as $x \\\\to -\\\\infty$, $P(x) \\\\to -\\\\infty$.\"],\"correctAnswer\":\"A\"}]},{\"id\":\"group-7\",\"questions\":[{\"id\":7,\"text\":\"Which of the following is the inverse of the function `$f(x) = 3^x$`?\",\"type\":\"single-choice\",\"options\":[\"$f^{-1}(x) = (\\\\frac{1}{3})^x$\",\"$f^{-1}(x) = \\\\log_3(x)$\",\"$f^{-1}(x) = x^3$\",\"$f^{-1}(x) = 3^{-x}$\"],\"correctAnswer\":\"B\"}]},{\"id\":\"group-8\",\"questions\":[{\"id\":8,\"text\":\"For the rational function `$g(x) = \\\\frac{x + 4}{x^2 - 16}$`, what is the domain?\",\"type\":\"single-choice\",\"options\":[\"All real numbers except $x = 4$.\",\"All real numbers except $x = -4$.\",\"All real numbers except $x = 4$ and $x = -4$.\",\"All real numbers.\"],\"correctAnswer\":\"C\"}]},{\"id\":\"group-9\",\"questions\":[{\"id\":9,\"text\":\"What is the center and radius of the circle given by the equation `$x^2 + y^2 - 6x + 8y - 11 = 0$`?\",\"type\":\"single-choice\",\"options\":[\"Center: $(3, -4)$, Radius: $6$\",\"Center: $(-3, 4)$, Radius: $6$\",\"Center: $(3, -4)$, Radius: $\\\\sqrt{11}$\",\"Center: $(-3, 4)$, Radius: $\\\\sqrt{11}$\"],\"correctAnswer\":\"A\"}]},{\"id\":\"group-10\",\"questions\":[{\"id\":10,\"text\":\"Which of the following is equivalent to `$\\\\tan(\\\\theta)
Exam Center
?\",\"type\":\"single-choice\",\"options\":[\"$\\\\frac{1}{\\\\sin(\\\\theta)}$\",\"$\\\\frac{1}{\\\\cos(\\\\theta)}$\",\"$\\\\sin(\\\\theta)\\\\cos(\\\\theta)$\",\"$\\\\frac{\\\\sin(\\\\theta)}{\\\\cos(\\\\theta)}$\"],\"correctAnswer\":\"D\"}]},{\"id\":\"group-11\",\"questions\":[{\"id\":11,\"text\":\"Consider the polynomial function `$P(x) = x^3 - 4x^2 - 11x + 30$`.\\n\\nGraph this function by finding its x-intercepts (roots), y-intercept, and describing its end behavior. Sketch the graph, clearly labeling the intercepts.\",\"type\":\"short-answer\",\"correctAnswer\":\"To graph $P(x) = x^3 - 4x^2 - 11x + 30$:\\n\\n1. **Find the roots (x-intercepts):**\\n * By Rational Root Theorem, possible rational roots are $\\\\pm 1, \\\\pm 2, \\\\pm 3, \\\\pm 5, \\\\pm 6, \\\\pm 10, \\\\pm 15, \\\\pm 30$.\\n * Test $x = 2$: $P(2) = (2)^3 - 4(2)^2 - 11(2) + 30 = 8 - 16 - 22 + 30 = 0$. So $x=2$ is a root.\\n * Use synthetic division with root 2:\\n ```\\n 2 | 1 -4 -11 30\\n | 2 -4 -30\\n -----------------\\n 1 -2 -15 0\\n ```\\n * The depressed polynomial is $x^2 - 2x - 15$. Factor this: $(x - 5)(x + 3)$.\\n * The roots are $x = 2, x = 5, x = -3$.\\n\\n2. **Find the y-intercept:**\\n * Set $x = 0$: $P(0) = (0)^3 - 4(0)^2 - 11(0) + 30 = 30$.\\n * The y-intercept is $(0, 30)$.\\n\\n3. **Determine End Behavior:**\\n * The leading term is $x^3$. This is an odd degree and positive leading coefficient.\\n * As $x \\\\to \\\\infty$, $P(x) \\\\to \\\\infty$.\\n * As $x \\\\to -\\\\infty$, $P(x) \\\\to -\\\\infty$.\\n\\n4. **Sketch the graph:**\\n * Plot the x-intercepts $(-3, 0), (2, 0), (5, 0)$.\\n * Plot the y-intercept $(0, 30)$.\\n * Use the end behavior: graph starts low on the left, goes up to cross $(-3,0)$, turns, goes down to cross $(2,0)$, turns again, and goes up through $(5,0)$ to positive infinity.\"}]},{\"id\":\"group-12\",\"questions\":[{\"id\":12,\"text\":\"Solve the logarithmic equation for $x$: `$\\\\log_5(x) + \\\\log_5(x - 4) = 1$`.\\n\\nShow all steps and check for extraneous solutions.\",\"type\":\"short-answer\",\"correctAnswer\":\"Given the equation: `$\\\\log_5(x) + \\\\log_5(x - 4) = 1$`\\n\\n1. **Use the logarithm property $\\\\log_b M + \\\\log_b N = \\\\log_b (MN)$:**\\n * `$\\\\log_5(x(x - 4)) = 1$`\\n * `$\\\\log_5(x^2 - 4x) = 1$`\\n\\n2. **Convert the logarithmic equation to an exponential equation ($\\\\log_b y = x \\\\iff b^x = y$):**\\n * `$5^1 = x^2 - 4x$`\\n * `$5 = x^2 - 4x$`\\n\\n3. **Rearrange into a quadratic equation:**\\n * `$x^2 - 4x - 5 = 0$`\\n\\n4. **Solve the quadratic equation (by factoring):**\\n * `$(x - 5)(x + 1) = 0$`\\n * Possible solutions: $x = 5$ or $x = -1$.\\n\\n5. **Check for extraneous solutions (the argument of a logarithm must be positive):**\\n * For $x = 5$: `$\\\\log_5(5) + \\\\log_5(5 - 4) = \\\\log_5(5) + \\\\log_5(1) = 1 + 0 = 1$`. This is a valid solution.\\n * For $x = -1$: `$\\\\log_5(-1)$` is undefined. Therefore, $x = -1$ is an extraneous solution.\\n\\n**The only valid solution is $x = 5$.**\"}]},{\"id\":\"group-13\",\"questions\":[{\"id\":13,\"text\":\"Subtract the following rational expressions and simplify your answer: `$\\\\frac{x}{x^2 - 4} - \\\\frac{2}{x + 2}$`.\\n\\nShow all steps.\",\"type\":\"short-answer\",\"correctAnswer\":\"To subtract the rational expressions: `$\\\\frac{x}{x^2 - 4} - \\\\frac{2}{x + 2}$`\\n\\n1. **Factor the denominators:**\\n * `$x^2 - 4 = (x - 2)(x + 2)$`\\n * The expression becomes: `$\\\\frac{x}{(x - 2)(x + 2)} - \\\\frac{2}{x + 2}$`\\n\\n2. **Find a common denominator:**\\n * The least common denominator (LCD) is $(x - 2)(x + 2)$.\\n * The second term needs to be multiplied by $\\\\frac{x - 2}{x - 2}$:\\n * `$\\\\frac{2}{x + 2} \\\\cdot \\\\frac{x - 2}{x - 2} = \\\\frac{2(x - 2)}{(x + 2)(x - 2)}$`\\n\\n3. **Rewrite the expression with the common denominator:**\\n * `$\\\\frac{x}{(x - 2)(x + 2)} - \\\\frac{2(x - 2)}{(x - 2)(x + 2)}$`\\n\\n4. **Combine the numerators:**\\n * `$\\\\frac{x - 2(x - 2)}{(x - 2)(x + 2)}$`\\n * `$\\\\frac{x - 2x + 4}{(x - 2)(x + 2)}$`\\n * `$\\\\frac{-x + 4}{(x - 2)(x + 2)}$`\\n\\n5. **Simplify the numerator (optional, but good practice):**\\n * `$\\\\frac{-(x - 4)}{(x - 2)(x + 2)}$`\\n\\n**The simplified expression is `$\\\\frac{-x + 4}{(x - 2)(x + 2)}$`.**\"}]},{\"id\":\"group-14\",\"questions\":[{\"id\":14,\"text\":\"Graph the parabola given by the equation `$y = x^2 - 4x + 3$`.\\n\\nClearly label the vertex and any intercepts.\",\"type\":\"short-answer\",\"correctAnswer\":\"To graph the parabola `$y = x^2 - 4x + 3$`:\\n\\n1. **Find the vertex:**\\n * The x-coordinate of the vertex is `$h = -\\\\frac{b}{2a} = -\\\\frac{-4}{2(1)} = \\\\frac{4}{2} = 2$`.\\n * The y-coordinate of the vertex is `$k = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1$`.\\n * Vertex: $(2, -1)$.\\n\\n2. **Find the y-intercept:**\\n * Set $x = 0$: $y = (0)^2 - 4(0) + 3 = 3$.\\n * Y-intercept: $(0, 3)$.\\n\\n3. **Find the x-intercepts (roots):**\\n * Set $y = 0$: `$x^2 - 4x + 3 = 0$`\\n * Factor: `$(x - 1)(x - 3) = 0$`\\n * X-intercepts: $(1, 0)$ and $(3, 0)$.\\n\\n4. **Sketch the graph:**\\n * Plot the vertex $(2, -1)$.\\n * Plot the y-intercept $(0, 3)$.\\n * Plot the x-intercepts $(1, 0)$ and $(3, 0)$.\\n * Since $a=1$ (positive), the parabola opens upwards.\\n * Draw a smooth U-shaped curve passing through these points.\\n\\n(A sketch showing a parabola opening upwards with vertex at (2,-1), crossing the x-axis at (1,0) and (3,0), and the y-axis at (0,3) would be required.)\"}]},{\"id\":\"group-15\",\"questions\":[{\"id\":15,\"text\":\"Solve the trigonometric equation `$2\\\\sin(\\\\theta) - 1 = 0$` for `$0 \\\\le \\\\theta \u003c 2\\\\pi$`.\\n\\nShow all steps.\",\"type\":\"short-answer\",\"correctAnswer\":\"To solve the trigonometric equation `$2\\\\sin(\\\\theta) - 1 = 0$` for `$0 \\\\le \\\\theta \u003c 2\\\\pi$`:\\n\\n1. **Isolate $\\\\sin(\\\\theta)$:**\\n * `$2\\\\sin(\\\\theta) = 1$`\\n * `$\\\\sin(\\\\theta) = \\\\frac{1}{2}$`\\n\\n2. **Identify angles on the unit circle where $\\\\sin(\\\\theta) = \\\\frac{1}{2}$:**\\n * In Quadrant I, the reference angle is `$\\\\theta = \\\\frac{\\\\pi}{6}$` (or $30^{\\\\circ}$). This is one solution.\\n * Sine is also positive in Quadrant II. The angle in Quadrant II with a reference angle of `$\\\\frac{\\\\pi}{6}$` is `$\\\\pi - \\\\frac{\\\\pi}{6} = \\\\frac{6\\\\pi - \\\\pi}{6} = \\\\frac{5\\\\pi}{6}$`. This is the second solution.\\n\\n**The solutions in the interval `$0 \\\\le \\\\theta \u003c 2\\\\pi$` are `$\\\\theta = \\\\frac{\\\\pi}{6}$` and `$\\\\theta = \\\\frac{5\\\\pi}{6}$`.**\"}]},{\"id\":\"group-16\",\"questions\":[{\"id\":16,\"text\":\"When the polynomial `$P(x) = 3x^3 - 2x^2 + x - 5$` is divided by `$(x + 1)
Exam Center
, what is the remainder?\",\"type\":\"single-choice\",\"options\":[\"$7$\",\"$1$\",\"$-11$\",\"$-5$\"],\"correctAnswer\":\"C\"}]},{\"id\":\"group-17\",\"questions\":[{\"id\":17,\"text\":\"Using a calculator, evaluate `$\\\\log_7(50)$` to four decimal places.\",\"type\":\"single-choice\",\"options\":[\"$1.9945$\",\"$0.5014$\",\"$1.7782$\",\"$0.6990$\"],\"correctAnswer\":\"A\"}]},{\"id\":\"group-18\",\"questions\":[{\"id\":18,\"text\":\"Solve the equation `$\\\\frac{3}{x} + \\\\frac{1}{2} = \\\\frac{x + 1}{x}$`.\",\"type\":\"single-choice\",\"options\":[\"$x = -2$\",\"$x = 4$\",\"$x = -4$\",\"No solution\"],\"correctAnswer\":\"B\"}]},{\"id\":\"group-19\",\"questions\":[{\"id\":19,\"text\":\"A parabolic satellite dish has a cross-section modeled by the equation `$y = \\\\frac{1}{16}x^2$`. If the receiver is placed at the focus of the parabola, how far is the receiver from the vertex?\",\"type\":\"single-choice\",\"options\":[\"$2$ units\",\"$4$ units\",\"$8$ units\",\"$16$ units\"],\"correctAnswer\":\"D\"}]},{\"id\":\"group-20\",\"questions\":[{\"id\":20,\"text\":\"A ladder is leaning against a wall. The ladder is $15$ feet long and forms an angle of $65^{\\\\circ}$ with the ground. How high up the wall does the ladder reach? Round to the nearest tenth of a foot.\",\"type\":\"single-choice\",\"options\":[\"$13.6$ feet\",\"$6.3$ feet\",\"$7.5$ feet\",\"$10.0$ feet\"],\"correctAnswer\":\"A\"}]},{\"id\":\"group-21\",\"questions\":[{\"id\":21,\"text\":\"Which of the following is a root of the polynomial `$P(x) = x^4 - 6x^3 + 12x^2 - 10x + 3$`?\",\"type\":\"single-choice\",\"options\":[\"$1$\",\"$2$\",\"$-1$\",\"$-3$\"],\"correctAnswer\":\"A\"}]},{\"id\":\"group-22\",\"questions\":[{\"id\":22,\"text\":\"The population of a certain bacteria doubles every $3$ hours. If you start with $100$ bacteria, how long will it take for the population to reach $10,000$ bacteria? Use the formula `$N(t) = N_0 \\\\cdot 2^{t/d}$` where $d$ is the doubling time. Round to the nearest hour.\",\"type\":\"single-choice\",\"options\":[\"$15$ hours\",\"$20$ hours\",\"$25$ hours\",\"$30$ hours\"],\"correctAnswer\":\"B\"}]},{\"id\":\"group-23\",\"questions\":[{\"id\":23,\"text\":\"What are the equations of the vertical and horizontal asymptotes for the function `$f(x) = \\\\frac{2x^2 - 5x + 3}{x^2 - 4}$`?\",\"type\":\"single-choice\",\"options\":[\"VA: $x = \\\\pm 2$, HA: $y = 0$\",\"VA: $x = 0$, HA: $y = 2$\",\"VA: $x = 2$, HA: $y = 2$\",\"VA: $x = \\\\pm 2$, HA: $y = 2$\"],\"correctAnswer\":\"D\"}]},{\"id\":\"group-24\",\"questions\":[{\"id\":24,\"text\":\"The equation `$x^2 + y^2 - 4x + 6y - 3 = 0$` represents a circle. What are its center and radius?\",\"type\":\"single-choice\",\"options\":[\"Center $(2, -3)$, Radius $4$\",\"Center $(-2, 3)$, Radius $4$\",\"Center $(2, -3)$, Radius $16$\",\"Center $(-2, 3)$, Radius $16$\"],\"correctAnswer\":\"A\"}]},{\"id\":\"group-25\",\"questions\":[{\"id\":25,\"text\":\"What is the amplitude and period of the function `$f(x) = -3\\\\cos(2x + \\\\pi) + 1$`?\",\"type\":\"single-choice\",\"options\":[\"Amplitude: $3$, Period: $2\\\\pi$\",\"Amplitude: $3$, Period: $\\\\pi/2$\",\"Amplitude: $3$, Period: $\\\\pi$\",\"Amplitude: $-3$, Period: $\\\\pi$\"],\"correctAnswer\":\"C\"}]},{\"id\":\"group-26\",\"questions\":[{\"id\":26,\"text\":\"Analyze the polynomial function `$f(x) = x^4 - x^3 - 7x^2 + x + 6$` by finding all its real roots, its y-intercept, and describing its end behavior. Then, sketch a detailed graph of the function, clearly labeling all intercepts.\",\"type\":\"short-answer\",\"correctAnswer\":\"To analyze and sketch the graph of `$f(x) = x^4 - x^3 - 7x^2 + x + 6$`:\\n\\n1. **Find the roots (x-intercepts) using Rational Root Theorem and Synthetic Division:**\\n * Possible rational roots: $\\\\pm 1, \\\\pm 2, \\\\pm 3, \\\\pm 6$.\\n * Test $x = 1$: $1 - 1 - 7 + 1 + 6 = 0$. So $x=1$ is a root.\\n ```\\n 1 | 1 -1 -7 1 6\\n | 1 0 -7 -6\\n ---------------------\\n 1 0 -7 -6 0\\n ```\\n * The depressed polynomial is $x^3 - 7x - 6$.\\n * Test $x = -1$: $(-1)^3 - 7(-1) - 6 = -1 + 7 - 6 = 0$. So $x=-1$ is a root.\\n ```\\n -1 | 1 0 -7 -6\\n | -1 1 6\\n -----------------\\n 1 -1 -6 0\\n ```\\n * The depressed polynomial is $x^2 - x - 6$.\\n * Factor $x^2 - x - 6 = (x - 3)(x + 2)$.\\n * The roots are $x = 1, x = -1, x = 3, x = -2$.\\n\\n2. **Find the y-intercept:**\\n * Set $x = 0$: $f(0) = (0)^4 - (0)^3 - 7(0)^2 + (0) + 6 = 6$.\\n * Y-intercept: $(0, 6)$.\\n\\n3. **Determine End Behavior:**\\n * The leading term is $x^4$. This is an even degree and positive leading coefficient.\\n * As $x \\\\to \\\\infty$, $f(x) \\\\to \\\\infty$.\\n * As $x \\\\to -\\\\infty$, $f(x) \\\\to \\\\infty$.\\n\\n4. **Sketch the graph:**\\n * Plot the x-intercepts: $(-2, 0), (-1, 0), (1, 0), (3, 0)$.\\n * Plot the y-intercept: $(0, 6)$.\\n * Use the end behavior: graph starts high on the left, goes down to cross $(-2,0)$, turns up, crosses $(-1,0)$, goes up to $(0,6)$, turns down, crosses $(1,0)$, turns, goes down to cross $(3,0)$, and then goes up to positive infinity.\\n\\n(A sketch showing the graph with these intercepts and correct end behavior would be provided.)\"}]},{\"id\":\"group-27\",\"questions\":[{\"id\":27,\"text\":\"Solve the equation `$e^{2x} - 5e^x + 6 = 0$` for $x$. Round your answers to three decimal places where appropriate.\\n\\nShow all steps.\",\"type\":\"short-answer\",\"correctAnswer\":\"To solve the equation `$e^{2x} - 5e^x + 6 = 0$`:\\n\\n1. **Recognize the quadratic form:** Let $u = e^x$. Then $e^{2x} = (e^x)^2 = u^2$.\\n * The equation becomes: `$u^2 - 5u + 6 = 0$`\\n\\n2. **Solve the quadratic equation for $u$ (by factoring):**\\n * `$(u - 2)(u - 3) = 0$`\\n * Possible values for $u$: $u = 2$ or $u = 3$.\\n\\n3. **Substitute back $e^x$ for $u$ and solve for $x$:**\\n * Case 1: `$e^x = 2$`\\n * Take the natural logarithm of both sides: `$\\\\ln(e^x) = \\\\ln(2)$`\\n * `$x = \\\\ln(2)$` (approximately $0.693$)\\n * Case 2: `$e^x = 3$`\\n * Take the natural logarithm of both sides: `$\\\\ln(e^x) = \\\\ln(3)$`\\n * `$x = \\\\ln(3)$` (approximately $1.099$)\\n\\n**The solutions are $x = \\\\ln(2)$ and $x = \\\\ln(3)$.**\"}]},{\"id\":\"group-28\",\"questions\":[{\"id\":28,\"text\":\"Analyze the rational function `$f(x) = \\\\frac{x^2 - 9}{x^2 - x - 6}$`.\\n\\nIdentify all holes, vertical asymptotes, horizontal asymptotes, x-intercepts, and y-intercepts. Then, sketch a detailed graph of the function, clearly labeling all these features.\",\"type\":\"short-answer\",\"correctAnswer\":\"To analyze and sketch the graph of `$f(x) = \\\\frac{x^2 - 9}{x^2 - x - 6}$`:\\n\\n1. **Factor numerator and denominator:**\\n * Numerator: `$x^2 - 9 = (x - 3)(x + 3)$`\\n * Denominator: `$x^2 - x - 6 = (x - 3)(x + 2)$`\\n * So, `$f(x) = \\\\frac{(x - 3)(x + 3)}{(x - 3)(x + 2)}$`\\n\\n2. **Identify Holes:**\\n * Since $(x - 3)$ is a common factor, there is a hole at $x = 3$.\\n * To find the y-coordinate of the hole, substitute $x = 3$ into the simplified function `$f(x) = \\\\frac{x + 3}{x + 2}$` (for $x \\\\ne 3$):\\n * `$y = \\\\frac{3 + 3}{3 + 2} = \\\\frac{6}{5}$`\\n * Hole at $(3, \\\\frac{6}{5})$.\\n\\n3. **Identify Vertical Asymptotes (VA):**\\n * Set the remaining denominator factor to zero: `$x + 2 = 0 \\\\implies x = -2$`.\\n * VA: $x = -2$.\\n\\n4. **Identify Horizontal Asymptotes (HA):**\\n * The degrees of the numerator ($2$) and denominator ($2$) are equal. The HA is the ratio of the leading coefficients.\\n * HA: `$y = \\\\frac{1}{1} = 1$`.\\n\\n5. **Find Intercepts:**\\n * **x-intercepts:** Set numerator (of simplified function) to zero: `$x + 3 = 0 \\\\implies x = -3$`.\\n * x-intercept: $(-3, 0)$.\\n * **y-intercept:** Set $x = 0$ in the simplified function:\\n * `$f(0) = \\\\frac{0 + 3}{0 + 2} = \\\\frac{3}{2}$`\\n * y-intercept: $(0, \\\\frac{3}{2})$.\\n\\n6. **Sketch the graph:**\\n * Draw the vertical asymptote $x = -2$ and horizontal asymptote $y = 1$.\\n * Plot the x-intercept $(-3, 0)$ and y-intercept $(0, \\\\frac{3}{2})$.\\n * Plot the hole at $(3, \\\\frac{6}{5})$.\\n * Sketch the branches of the hyperbola, respecting the asymptotes and passing through the intercepts and the hole. Note that the graph will approach the asymptotes but not cross them (for the VA) or only cross them in the middle for the HA for rational functions.\\n\\n(A detailed sketch showing the asymptotes, intercepts, and the hole with the appropriate curve branches would be required.)\"}]},{\"id\":\"group-29\",\"questions\":[{\"id\":29,\"text\":\"Consider the equation `$4x^2 + 9y^2 = 36$`.\\n\\nIdentify the type of conic section, convert it to standard form, find its center, vertices, and foci. Then, sketch the graph, clearly labeling the center, vertices, and foci.\",\"type\":\"short-answer\",\"correctAnswer\":\"To graph the ellipse `$4x^2 + 9y^2 = 36$` and find its foci:\\n\\n1. **Convert to standard form:** Divide the entire equation by $36$.\\n * `$\\\\frac{4x^2}{36} + \\\\frac{9y^2}{36} = \\\\frac{36}{36}$`\\n * `$\\\\frac{x^2}{9} + \\\\frac{y^2}{4} = 1$`\\n\\n2. **Identify $a^2$ and $b^2$:**\\n * Since $9 \u003e 4$, $a^2 = 9$ and $b^2 = 4$.\\n * So, $a = \\\\sqrt{9} = 3$ and $b = \\\\sqrt{4} = 2$.\\n\\n3. **Determine the orientation and vertices:**\\n * Since $a^2$ is under $x^2$, the major axis is horizontal.\\n * Center: $(0, 0)$ (as there are no $h$ or $k$ terms).\\n * Vertices: $(\\\\pm a, 0) = (\\\\pm 3, 0)$.\\n * Co-vertices: $(0, \\\\pm b) = (0, \\\\pm 2)$.\\n\\n4. **Calculate $c$ to find the foci:**\\n * For an ellipse, `$c^2 = a^2 - b^2$`\\n * `$c^2 = 9 - 4 = 5$`\\n * `$c = \\\\sqrt{5}$` (approximately $2.236$)\\n\\n5. **Locate the foci:**\\n * Since the major axis is horizontal, the foci are at $(\\\\pm c, 0)$.\\n * Foci: $(\\\\pm\\\\sqrt{5}, 0)$.\\n\\n6. **Sketch the graph:**\\n * Plot the center $(0,0)$.\\n * Plot the vertices $(\\\\pm 3, 0)$ and co-vertices $(0, \\\\pm 2)$.\\n * Draw a smooth elliptical curve through these points.\\n * Mark the foci $(\\\\pm\\\\sqrt{5}, 0)$.\\n\\n(A sketch showing an ellipse centered at the origin, with x-intercepts at $\\\\pm 3$, y-intercepts at $\\\\pm 2$, and foci marked at $(\\\\pm\\\\sqrt{5}, 0)$ would be required.)\"}]},{\"id\":\"group-30\",\"questions\":[{\"id\":30,\"text\":\"Graph the trigonometric function `$y = 2\\\\sin(2x - \\\\pi) + 1$`.\\n\\nIdentify the amplitude, period, phase shift, and vertical shift. Show at least one full period of the graph, clearly labeling key points (e.g., maximums, minimums, and intercepts or points on the midline).\",\"type\":\"short-answer\",\"correctAnswer\":\"To graph the function `$y = 2\\\\sin(2x - \\\\pi) + 1$`:\\n\\n1. **Rewrite in standard transformation form:** `$y = 2\\\\sin(2(x - \\\\frac{\\\\pi}{2})) + 1$`\\n\\n2. **Identify key features:**\\n * **Amplitude (A):** $|A| = |2| = 2$. The graph oscillates 2 units above and below the midline.\\n * **Period (P):** `$P = \\\\frac{2\\\\pi}{|B|} = \\\\frac{2\\\\pi}{2} = \\\\pi$`. One full cycle completes in $\\\\pi$ units.\\n * **Phase Shift (C):** `$C = \\\\frac{\\\\pi}{2}$` to the right. The starting point of a cycle is shifted to $x = \\\\frac{\\\\pi}{2}$.\\n * **Vertical Shift (D):** $D = 1$. The midline of the graph is $y = 1$.\\n\\n3. **Determine the interval for one cycle:**\\n * The basic sine wave starts at $0$. With the phase shift, the new starting point is `$x = \\\\frac{\\\\pi}{2}$`.\\n * The cycle ends at `$\\\\frac{\\\\pi}{2} + P = \\\\frac{\\\\pi}{2} + \\\\pi = \\\\frac{3\\\\pi}{2}$`.\\n * Interval for one cycle: `$[\\\\frac{\\\\pi}{2}, \\\\frac{3\\\\pi}{2}]$`.\\n\\n4. **Find five key points within one cycle:**\\n * Divide the period by 4: `$\\\\frac{\\\\pi}{4}$`.\\n * Start at $x = \\\\frac{\\\\pi}{2} = \\\\frac{2\\\\pi}{4}$. Add increments of `$\\\\frac{\\\\pi}{4}$`:\\n * $x_1 = \\\\frac{2\\\\pi}{4} = \\\\frac{\\\\pi}{2}$ (Start of cycle, on midline)\\n * $x_2 = \\\\frac{2\\\\pi}{4} + \\\\frac{\\\\pi}{4} = \\\\frac{3\\\\pi}{4}$ (Maximum)\\n * $x_3 = \\\\frac{3\\\\pi}{4} + \\\\frac{\\\\pi}{4} = \\\\frac{4\\\\pi}{4} = \\\\pi$ (Midline)\\n * $x_4 = \\\\pi + \\\\frac{\\\\pi}{4} = \\\\frac{5\\\\pi}{4}$ (Minimum)\\n * $x_5 = \\\\frac{5\\\\pi}{4} + \\\\frac{\\\\pi}{4} = \\\\frac{6\\\\pi}{4} = \\\\frac{3\\\\pi}{2}$ (End of cycle, on midline)\\n\\n * Corresponding y-values (for `$y = 2\\\\sin(2x - \\\\pi) + 1$`):\\n * At $x = \\\\frac{\\\\pi}{2}$: $y = 2\\\\sin(0) + 1 = 1$\\n * At $x = \\\\frac{3\\\\pi}{4}$: $y = 2\\\\sin(\\\\frac{\\\\pi}{2}) + 1 = 2(1) + 1 = 3$\\n * At $x = \\\\pi$: $y = 2\\\\sin(\\\\pi) + 1 = 2(0) + 1 = 1$\\n * At $x = \\\\frac{5\\\\pi}{4}$: $y = 2\\\\sin(\\\\frac{3\\\\pi}{2}) + 1 = 2(-1) + 1 = -1$\\n * At $x = \\\\frac{3\\\\pi}{2}$: $y = 2\\\\sin(2\\\\pi) + 1 = 2(0) + 1 = 1$\\n\\n * Key points: $(\\\\frac{\\\\pi}{2}, 1)$, $(\\\\frac{3\\\\pi}{4}, 3)$, $(\\\\pi, 1)$, $(\\\\frac{5\\\\pi}{4}, -1)$, $(\\\\frac{3\\\\pi}{2}, 1)$.\\n\\n5. **Sketch the graph:**\\n * Draw the midline $y = 1$.\\n * Plot the five key points within one cycle.\\n * Connect the points with a smooth sinusoidal curve.\\n * Extend the graph in both directions if desired to show more cycles.\\n\\n(A sketch showing a sine wave with amplitude 2, period $\\\\pi$, shifted right by $\\\\frac{\\\\pi}{2}$, and up by 1, passing through the key points identified would be required.)\"}]}],\"targetAudience\":\"Grade 11 Students\",\"creationTime\":1759340414765,\"lastVisited\":1759340414765,\"model\":\"predefined\",\"timeTaken\":0,\"apiProvider\":\"static\"}},\"actionData\":null,\"errors\":null}");