Question

circuit training - precalculus unit 1 review! name directions: begin in cell #1. show the work necessary to answer the question. circle your answer. search for the answer, call that cell #2 and proceed in this manner until you complete the circuit. do not hesitate to use separate paper to showcase your best work. note: calculator use is permitted on questions 14, 15, and 16 only. # 1 answer: -2.079 the graph of f(x) is shown. at what value of x is f(x) both increasing and concave down? -5, -3, 1, 2, 5 # answer: 0 functions f and g are defined for all real numbers. in the x - y plane, the graph of g is constructed by applying three transformations to the graph of f in this order: a horizontal dilation by a factor of \\(\frac{2}{3}\\), a horizontal translation of 9 units, and a vertical translation of 2 units. g(x)=f(a(x + h))+k. fill in the blanks: a = h = k = to advance in the circuit, search for a hk: # answer: -5 at what x - value does the rational function y = \\(\frac{2x + 6}{x^{2}-9}\\) have a hole? # answer: 5 \\(\lim_{x\to\infty}\frac{4x - 3}{3 + x}=?\\) # answer: 4 the function p(x)=\\(\frac{5x^{3}+4x^{2}-3x + 7}{2x^{2}-4}\\) has a slant asymptote of the form y = ax + b, where a = and b = to advance in the circuit, find the sum of a and b. # answer: 1 circle all of the asymptotes for the graph of the function h(x)=\\(\frac{x^{2}+3x + 2}{x^{2}-x}\\). x = 0, x = 1, y = 0, y = 1, y = 1x + 0 how many did you circle? # answer: 6229 the function v gives the velocity in meters per second of a particle in an accelerator machine. what is the value of v(7) predicted by the quartic function model? t 0 1 2 3 4 v(t) 2 5 34 125 338 # answer: 2 \\(\lim_{x\to-\infty}-\frac{1}{2}x^{n}+3x - 4=\infty\\) if the above is true, n could equal which two of these choices? 2, 3, 4, 5 search for the largest value you circled. # answer: \\(\frac{9}{2}\\) what is the average rate of change of the function h(x)=x^{2}-x - 1 on the interval -5,1? # answer: -3 how many vertical asymptotes does the function g(x)=\\(\frac{2x^{2}-5x - 3}{x^{2}-4x + 3}\\) have? # answer: 3 what is the x - coordinate for the point of inflection for the graph of y=-x(x - 2)(x + 2)? # answer: 2669 determine the absolute maximum value for the graph of f(x)=-0.22x^{4}+3x^{2}-4.05x - 7 on the interval -2\\(\leq\\)x\\(\leq\\)3. © virge cornelius 2023. made by request for c. simpson, her lhs students and precalculus aficionados everywhere!

Explanation:

Step1: Analyze increasing and concave - down of $f(x)$

Inspect the graph of $f(x)$ to find where the slope is positive (increasing) and the second - derivative is negative (concave down).

Step2: Determine transformation parameters for $g(x)$

For a horizontal dilation by a factor of $\frac{2}{3}$, $a=\frac{3}{2}$. For a horizontal translation of 9 units to the right, $h=- 9$. For a vertical translation of 2 units up, $k = 2$.

Step3: Find the hole of the rational function

Factor $y=\frac{2x + 6}{x^{2}-9}=\frac{2(x + 3)}{(x + 3)(x - 3)}$. The hole is at $x=-3$.

Step4: Calculate the limit $\lim_{x

ightarrow\infty}\frac{4x-3}{3 + x}$
Divide both numerator and denominator by $x$: $\lim_{x
ightarrow\infty}\frac{4-\frac{3}{x}}{\frac{3}{x}+1}=4$.

Step5: Find slant asymptote of $p(x)$

Use long - division or synthetic division on $p(x)=\frac{5x^{3}+4x^{2}-3x + 7}{2x^{2}-4}$. $a=\frac{5}{2}$, $b = 2$, sum $a + b=\frac{5}{2}+2=\frac{9}{2}$.

Step6: Identify asymptotes of $h(x)$

Factor $h(x)=\frac{x^{2}+3x + 2}{x^{2}-x}=\frac{(x + 1)(x + 2)}{x(x - 1)}$. Vertical asymptotes at $x = 0$ and $x = 1$, horizontal asymptote at $y = 1$. Circle 3 asymptotes.

Step7: Predict $V(7)$ using quartic model

Assume a quartic model $V(t)=at^{4}+bt^{3}+ct^{2}+dt+e$. Use the given data points to find the coefficients and then substitute $t = 7$.

Step8: Determine $n$ for $\lim_{x

ightarrow-\infty}-\frac{1}{2}x^{n}+3x - 4=\infty$
For the limit to be $\infty$ as $x
ightarrow-\infty$, $n$ must be even. So $n = 2$ or $n = 4$.

Step9: Calculate average rate of change of $h(x)$

The average rate of change of $h(x)=x^{2}-x - 1$ on $[-5,1]$ is $\frac{h(1)-h(-5)}{1-(-5)}=\frac{(1^{2}-1 - 1)-((-5)^{2}-(-5)-1)}{6}=\frac{-1-(25 + 5-1)}{6}=\frac{-1-29}{6}=-5$.

Step10: Find vertical asymptotes of $g(x)$

Factor $g(x)=\frac{2x^{2}-5x - 3}{x^{2}-4x + 3}=\frac{(2x + 1)(x - 3)}{(x - 1)(x - 3)}$. After canceling out the common factor, the vertical asymptote is at $x = 1$. There is 1 vertical asymptote.

Step11: Find $x$ - coordinate of inflection point of $y=-x(x - 2)(x + 2)=-x^{3}+4x$

Find the second - derivative $y'=-3x^{2}+4$, $y''=-6x$. Set $y'' = 0$, $x = 0$.

Step12: Find absolute maximum of $f(x)=-0.22x^{4}+3x^{2}-4.05x - 7$ on $[-2,3]$

Find the critical points by setting $f'(x)=-0.88x^{3}+6x-4.05 = 0$ and evaluate $f(x)$ at critical points and endpoints.

Answer:

The steps above show the work for each part of the circuit - training problem. The specific answers for each blank in the circuit are found through the calculations in each step. For example, for the transformation of $g(x)=f(a(x + h))+k$, $a=\frac{3}{2}$, $h=-9$, $k = 2$; for the limit $\lim_{x
ightarrow\infty}\frac{4x-3}{3 + x}=4$; etc.