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Grade 11 Nonlinear Equations and Functions CCSS: HSA-REI.A.2
Grade 11 Nonlinear Equations and Functions CCSS: HSA-REI.A.2
Solve for all possible values of $x$. $sqrt(x^2+4x-16)=4$
1. No Solution
2. $-8, 4$
3. All real numbers
4. $-4, 16$
Grade 11 Nonlinear Equations and Functions CCSS: HSA-REI.A.2
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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After some more algebra, and then moving all terms with $x$ or $y$ to the left side of the equation (and all others to the right), the resulting equation is:

 $a^2 x^2 - c^2 x^2 + a^2 y^2$ $=$ $a^4 -a^2 c^2$ $(a^2 - c^2)x^2 + a^2 y^2$ $=$ $(a^2 - c^2) a^2$ $b^2 x^2 + a^2 y^2$ $=$ $a^2 b^2$ $x^2 / a^2 + y^2 / b^2$ $=$ $1$

This is the formula for the equation of an ellipse centered at $(0,0)$. Why can the substitution $b^2 = a^2-c^2$ be made?
1. Since $b$ is undefined so far, it can be defined as any value. Then, using the Pythagorean theorem, $a^2 + b^2 = c^2$, simply rearrange to solve for $b^2$.
2. Knowing that the semi-minor axis is $b$ units long, one can substitute the square of this value for $a^2-c^2$.
3. Since $b$ is not yet defined, it can be used to simplify the equation by defining $b^2 = a^2 - c^2$. A positive value for $a^2-c^2$ exists since $|a| > |c|$.
4. Projecting $b$, the length of the semi-minor axis, onto the semi-major axis it is seen that $b=a-c$. Then, simply square both sides of the equation.
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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After some more algebra, the equation becomes:
$(c^2-a^2) x^2 - y^2 = (c^2 - a^2) y^2$
What happened to the plus-minus sign that was in the original equation in the seventh question?
1. Since there are no more square root terms, it is unnecessary.
2. Each term with it was eventually squared, and a squared term must be positive it.
3. Because the terms were moved from one side to another multiple times, it is no longer necessary.
4. Any terms with it canceled out.
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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Substituting the equations for $d_1, d_2$ found in the second and third questions into the above equation, and then moving the second square root term to the right side of the equation results in the following:
$sqrt( (x+c)^2 + y^2) = 2a - sqrt( (x-c)^2 + y^2)$
The next step will be to square both sides. Why was the second square root term moved to the right side of the equation before squaring, instead of just squaring both sides immediately?
1. It gets rid of all square root terms immediately.
2. It will make the algebra easier later on, since the two square root terms are not multiplied together.
3. This ensures that there won't be any multiple answers that usually result from square roots (taking the positive and negative).
4. This makes that equation more balanced, which looks nicer.
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.1

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Having found the equation of the circle centered at $(3,4)$ with radius of length 2 in the previous question, which of the following best describes the meaning of this equation?
1. There are always two points inside a circle with which to form a right triangle using the radius of the circle as the hypotenuse.
2. Any point (x,y) that is 2 units away from the center of this circle lies on the circle.
3. There exists one point, (x,y), which solves this equation.
4. All points that lie on this circle also must be part of a right triangle.
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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Letting $b^2 = c^2 - a^2$, the equation no becomes:
$b^2 x^2 - y^2 = b^2 a^2$
$x^2 / a^2 - y^2 / b^2 = 1$
This is the equation of a hyperbola, centered at the origin with foci $(-c,0)$ and $(c,0)$. Where does the substitution $b^2 = c^2 - a^2$ come from?
1. Since $b$ usually appears in the equation for a hyperbola, it must be included. Using the Pythagorean theorem, $a^2 +b^2 = c^2$, simply rearrange the equation.
2. It has to be done, to ensure the asymptotes are related to the equation. The equations of the asymptotes are $y= pm b/a$, and knowing that $|c| > |a|$, squaring and rearranging results in $b^2 = c^2 - a^2$.
3. It's done to simply the equation. $b$ is not defined yet, and since $|c| > |a|$, $c^2 > a^2$, and so there must be a positive number, $b^2$ such that $b^2 = c^2 - a^2$.
4. Knowing that $b$ is the length of the semi-minor axis, a right triangle can be formed with the center of the hyperbola and either foci, with $b$ as the length of one leg of this triangle. Applying the Pythagorean theorem results in $b^2 + c^2 = a^2$, and simply rearrange.
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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Which equation represents the distance, $d_2$, between $F_2$ and the point $(x,y) ?$
1. $d_2 = sqrt( (c-x)^2 + (y-c)^2)$
2. $d_2 = sqrt( (x-c)^2 + y^2)$
3. $d_2 = sqrt( (c-x)^2)$
4. $d_2 = sqrt( (x+c)^2 + y^2)$
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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Let $(-a,0)$ be the left vertex of the ellipse. What is the distance from this vertex to $F_1 ?$
1. $sqrt((c-a)^2 + y^2)$
2. $sqrt(a^2 + c^2)$
3. $a$
4. $a-c$
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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What is the distance between $(-a,0)$ and $F_2 ?$
1. $a+c$
2. $c-a$
3. $a-c$
4. $sqrt(a^2 - c^2)$
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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Let $(x,y)$ be a point on the ellipse. Which equation represents the distance, $d_1$, between $F_1$ and the point $(x,y) ?$
1. $d_1 = sqrt( (x+c)^2 + y^2)$
2. $d_1 = sqrt( (x-c) + y^2)$
3. $d_1 = sqrt( (-x + c)^2 + y^2$
4. $d_1 = sqrt( (x-c)^2 + (y-x)^2)$
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.1

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For the points which are excluded by the condition mentioned in the previous question, does the equation of the circle not work for them? Why or why not?
1. It doesn't, because either the x or y distance becomes zero meaning an entirely different equation is required.
2. It doesn't, these points are always excluded, but just usually ignored.
3. It does, however these points would just have introduced complex numbers into the derivation and so were avoided.
4. It does, these points are simply special cases where the distance from the center to the circle is all in either the x or y direction.
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.1

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In the first question, the stipulation on the point $(x,y)$ was made that the x-coordinate could not be 3, and the y-coordinate could not be 4. Why was this?
1. These points do not actually occur on the circle, and so would have caused an error in the derivation.
2. If a point met either of these criteria, it would be either directly above, below, to the right of, or to the left of the center of the circle. This would have made creating a right triangle with legs that have slope of 0 and undefined impossible.
3. These points would have resulted in complex values (calculations involving imaginary numbers), and led to serious complications in the derivation process.
4. The points that meet these criteria result in triangles which are not right triangles.
Grade 11 Nonlinear Equations and Functions CCSS: HSA-SSE.B.3, HSA-SSE.B.3c
Does the function $y=4^(-2x)$ represent exponential growth or decay? What is the percent rate of change?
1. exponential growth; 4%
2. exponential decay; 93.75%
3. exponential growth; 83.5%
4. exponential decay; 6.25%
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.1

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If there were multiple correct answers in the previous question, will this result in different triangles?
1. No, they will be congruent (one will be a transformation of the other).
2. Yes, one will be larger than the other.
3. Yes, one will be an isosceles right triangle, the other will be a scalene right triangle.
4. No, there was only one right answer.
Grade 11 Nonlinear Equations and Functions CCSS: HSA-REI.A.2
Solve for $m$. $(2m)/(m+3) = 4$
1. $m=2$
2. $m = -6$
3. $m=3$
4. $m = 3/7$
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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Grade 11 Nonlinear Equations and Functions

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Squaring both sides, expanding, and combing like terms, dividing by any common constants, and putting all terms without a square root on the left side of the equation results in which of the following equations?
1. $2cx - a = 0$
2. $cx pm a^2 = -2 sqrt((x-c)^2 + y^2)$
3. $cx - a^2 = 0$
4. $cx - a^2 = pm a sqrt( (x-c)^2 + y^2)$
Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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Let $(x,y)$ be a point on the hyperbola. What is the distance, $d_1$, between $F_1$ and the point $(x,y) ?$
1. $d_1 = sqrt( (x+c)^2 + y^2)$
2. $d_1 = sqrt( (x-c)^2 + y^2)$
3. $d_1 = sqrt( (x^2 + y^2)$
4. $d_1 = sqrt( (x-c)^2 + (y-c)^2)$
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