Calculus
Limits, derivatives, optimization, and behavior of functions.
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3) sketch the graph of $f(x) = \\log_{3} x$. what is $b$ in this proble…
- $b=3$ - Completed table: | $x$ | $f(x)=\log_{3}x$ | |-----|-------------------| | $\frac{1}{3}$ | $-1$ | | $1$ | $0$ | | $3$ | $1$ | - The graph of $f(x)=\log_{3}x$ has a vertic…
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exercise 1 (answers on page 40) multiply these fractions. cancel and si…
$\boldsymbol{\frac{1}{12}}$ ### Problem 2: $\boldsymbol{\frac{1}{2} \times \frac{4}{5}}$
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use the figure below to answer the question that follows. the graph of …
D. \(g(x) = f(x - 1) + 2\)
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top heavy fractions may contain common factors as well. they will need …
\(6\frac{1}{2}\) ### 2. Solve \(\boldsymbol{\frac{40}{6}}\)
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to add further points: if you wish to add additional points to your gra…
To graph \(y = 2x-6\), plot the points \((0,-6)\) (y - intercept) and \((3,0)\) (x - intercept) and draw a straight line through them. The line has a slope of \(2\) and a y - inte…
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14. race response (use the race rubric to help you earn the whole 20 po…
*: Nichelle Nichols broke barriers on - and off - screen: she pioneered a groundbreaking role on *Star Trek* and advocated for diversity in STEM, especially at NASA, inspiring und…
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choose all that correctly graphs the function. a $f(x) = \\frac{1}{5}x …
A. Graph matching $f(x)=\frac{1}{5}x - 2$, B. Graph matching $f(x)=\frac{2}{3}x + 3$
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graph the following function: ( f(x)=-\frac{3}{4}x + 7 )
The line passes through points $(0, 7)$ and $(4, 4)$, extending infinitely in both directions on the coordinate grid.
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7. $f(x)=vert x + 2vert + 1$
The graph of \( f(x)=|x + 2|+1 \) is a V - shaped graph (absolute - value graph) with vertex at \( (-2,1) \), opening upwards. The key points are \( (-4,3) \), \( (-3,2) \), \( (-…
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graph the following equation on the coordinate plane: $6x + 3y = -12$. …
Plot the points $(-2, 0)$ and $(0, -4)$ on the coordinate plane, then draw a straight line through them to represent the equation $6x + 3y = -12$.
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identify the vertex, then graph 7. $f(x)=|x + 2|+1$ 8. $f(x)=-dfrac{5}{…
The vertex of \( f(x) = |x + 2| + 1 \) is \((-2, 1)\) ### Problem 8: \( f(x) = -\frac{5}{2}|x - 4| \)
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the graph of a linear function p is shown below. what is the y - coordi…
3
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use properties of logarithms to expand each logarithmic expression as m…
$1 + \log_{2}(x)$
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use the graphs and work shown below to answer the question that follows…
To solve for the sums of the areas of the rectangles (A and B) for the curve \( y = x^2 \) from \( x = 0 \) to \( x = 4 \), we analyze each case: ### Calculating Sum of Areas \( A…
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the graph of ( f ) is shown below. write an equation of the rational fu…
$R(x)=3-\frac{10}{x}$
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the graph of ( f ) is shown below. write an equation of the rational fu…
$R(x)=\frac{3x-2}{x-2}$
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the graph of ( f ) is shown below. write an equation of the rational fu…
$R(x)=\frac{-2}{x}+3$ or $R(x)=3-\frac{2}{x}$
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problem $y = \\frac{2 - x}{x + 2}$ x - intercept: 2 vertical asymptote:…
$- + -$
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problem $y = \\frac{2 - x}{x + 2}$ x-intercept: e.g., 1 degree: vertica…
x-intercept: $2$ degree (numerator): $1$ vertical asymptote: $x=-2$ degree (denominator): $1$ leading term: $\frac{-x}{x}$ (or $-1$) horizontal asymptote: $y=-1$ pattern of signs:…
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joaquins car has a full tank of gas at the start of a trip. the graph s…
10
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find an appropriate viewing window for the function, using the given x-…
1. Appropriate viewing window: (Note: The given options C/D are incomplete, but the only valid window that includes both positive and negative y-values for the key points would ne…
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look at the graph: what are the equations of the horizontal and vertica…
horizontal asymptote: \( y = 5 \) vertical asymptote: \( x = 0 \)
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find an appropriate viewing window for the function, using the given x …
First question: B. $[-15,15]$ by $[-90,190]$ Second question: D. (The graph that rises to the right, falls to the left, with a local maximum on the negative x-axis and local minim…
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answer the questions below about the quadratic function.\\( f(x) = -2x^…
(a) Maximum (b) $x=3$ (c) $2$
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find an appropriate viewing window for the function, using the given x-…
## Graph Selection: A. (The upward-opening parabola graph with lowest point at the center) ## Turning Point: A. The graph has a turning point in this window. The turning point(s) …
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find the difference quotient of f; that is find \\(\\frac{f(x + h) - f(…
\(\frac{1}{\sqrt{x + h - 6}+\sqrt{x - 6}}\)
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find the difference quotient of f, that is, find \\(\\frac{f(x + h) - f…
\( 2x + h - 9 \)
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suppose the cost of the production and sale of x units of an electronic…
C. <The upward-sloping graph shown in option C>
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use the graph to answer the question. describe the continuity or discon…
The function is continuous on the intervals $(-6, -2)$, $(-2, 0)$, and $(0, \infty)$. - At $x=-2$, the function has a removable discontinuity (the limit exists, but the function's…
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complete parts a. and b. below. | x | y | |----|----| | -3 | -29 | | -2…
1. The given table values are all correct for $y=x^3-2$. 2. The graph is a smooth cubic curve passing through the points $(-3,-29), (-2,-10), (-1,-3), (0,-2), (1,-1), (2,6), (3,25…
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here is a graph of a segment of $f(x) = x$. $g(x) = f(x - 1)$ graph $g$.
To graph \( g(x)=f(x - 1) \) (where \( f(x)=x \)): 1. Recognize that \( g(x) \) is \( f(x) \) shifted 1 unit to the right. 2. Take key points on \( f(x)=x \) (e.g., \( (-2,-2) \) …
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complete parts a. and b. below. b. graph the function $y = x^3 + 1$ wit…
B. <The graph of an increasing cubic function passing through (-1,0) and (0,1), curving upward for positive x and downward for negative x>
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7. for $f(x) = 4x - 1$, find $f(x)$ when $x = 4$. a. 12 b. 19 c. 15 d. …
c. 15
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6. for $f(x) = 9x - 6$, find $f(x)$ when $x = 4$. a. 30 b. -18 c. 39 d.…
a. 30
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5. for ( f(x) = -9x + 11 ), find ( f(x) ) when ( x = 5 ). a. 56 b. -34 …
b. -34
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pathfinder choose a path from start to finish below. use subtraction, m…
# Explanation: ## Step1: Calculate start cell $3 - 24 = -21$ ## Step2: Calculate row1, col2 $3 \times (-13) = -39$ ## Step3: Calculate row1, col3 $-3 \times 14 = -42$ ## Step4: Ca…
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what is the slope of this line? simplify your answer and write it as a …
2
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what is the slope of this line? simplify your answer and write it as a …
$\frac{-2}{3}$
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what is the slope of this line? simplify your answer and write it as a …
$\frac{3}{5}$
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what is the slope of this line? simplify your answer and write it as a …
$\frac{3}{2}$
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the polynomial function $f(x)$ is graphed below. fill in the form below…
The degree of \(f(x)\) is \(4\) (even) and the leading coefficient is negative. There are \(2\) different real zeros and \(1\) relative minimums. So filling in the blanks: The deg…
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question express in simplest radical form. \\(\\sqrt{100}\\)
10
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ch. 4 canvas quiz #1 this quiz is over our 4.3 notes from yesterday in …
A. \(y=\frac{2}{3}x - 8\)
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find the domain and range of the function shown in the graph given belo…
$-6, -4, -2, 1, 4, 7$
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find the slope of the line that passes through the pair of points. (4, …
# Explicación: ## Paso1: Identificar la fórmula de la pendiente La fórmula para la pendiente $m$ entre dos puntos $(x_1,y_1)$ y $(x_2,y_2)$ es: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$…
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quiz 2.1: graphs of polynomial functions score: 6/10 answered: 4/5 ques…
- The coordinates of its g - intercept are \((0, 42)\). - The coordinates of its t - intercepts are \((6, 0)\), \((-7, 0)\), \((1, 0)\)
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given the function $p(x) = x^3 + 5x^2 - 14x$. the $y$-intercept is $(0,…
s: - The y - intercept is \(\boldsymbol{(0,0)}\). - The x - intercepts are \(\boldsymbol{(-7,0),(0,0),(2,0)}\). - When \(x ightarrow\infty\), \(y ightarrow\boldsymbol{\infty}\). -…
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do the following for the function $f(x)=\frac{2}{x}$. (a) express the s…
$\frac{-2}{x(x+h)}$ ---
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given the function $p(x) = x^3 + 5x^2 - 14x$. the $y$-intercept is the …
The y - intercept is \(0\). ### x - intercepts
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write an equation for the polynomial graphed below y(x)=
\(\frac{1}{27}(x + 3)^{2}(x - 3)^{2}\) (or expanded form \(\frac{1}{27}x^{4}-\frac{2}{3}x^{2}+3\))
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what are the domain and range of $f(x) = -30$? a domain: all real numbe…
A. Domain: All real numbers Range: {-30}
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question #4 the authors of the constitution drew much of their inspirat…
False ### Question #5
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american government/civics unit 1: foundations of government unit 2: or…
To solve this matching problem, we analyze each principle and its definition: 1. **Rule of Law**: The definition "government officials, lawmakers, and even the president are not a…
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write the equation of this line in slope - intercept form. write your a…
$y = x + 4$
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write a linear equation using ( x ) and ( y ) for the line graphed belo…
\( x = 2 \)
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question solve for ( x ), rounding to the nearest hundredth. ( 16 cdot …
\(x\approx0.16\)
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question graph the function $f(x) = \\log_{2} x - 1$ on the axes below.…
The vertical asymptote is \( x = 0 \). Two points with integer coordinates can be \( (1,-1) \) and \( (2,0) \) (or other valid points like \( (1,-1) \) and \( (4,1) \) etc.). The …
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express the following rational number as a decimal. \\(\frac{4}{13}\\) …
A. 0.307692
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which graph represents the function $f(x)=\\frac{2}{x - 1}+4$?
The graph (the one with vertical asymptote \( x = 1 \) and horizontal asymptote \( y = 4 \), showing the correct behavior around \( x = 1 \) and \( y = 4 \)) represents the functi…
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which graph represents the function $f(x)=\frac{2}{x - 1}+4$?
The graph should have a vertical asymptote at \( x = 1 \), a horizontal asymptote at \( y = 4 \), be a transformation (shifted 1 unit right and 4 units up) of the function \( y=\f…
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which of the following terms does not appear in the complete partial fr…
# Explanation: To determine which term does not appear in the partial fraction decomposition, we analyze the factors of the denominator: ### Step 1: Analyze each factor - The deno…
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consider the integral \\(\\int \\sqrt{x^2 - 25} \\, dx\\) which of the …
B. Trigonometric substitution with \(x = 5\sec\theta\)
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determine the interval on which the function in the graph below is decr…
The function is decreasing on the interval \(\boxed{[-4, 6]}\)
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which of the following techniques would you use to solve the integral \…
B. u - substitution with \(u = x^{2}+7\)
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evaluate the indefinite integral. \\(\\int \\tan^{7}(x) \\sec^{4}(x) \\…
\(\frac{\tan^{10}(x)}{10}+\frac{\tan^8(x)}{8}\)
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what is the slope of the line that passes through the points (-4,5) and…
\(-1\)
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h(x) = -x^4 + 3x^3 - 2x^2
The factored form of \(h(x)=-x^4 + 3x^3 - 2x^2\) is \(\boldsymbol{-x^2(x - 1)(x - 2)}\)
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find the following indefinite integral. \\(\\int \\frac{3t^2}{(5t^3 + 8…
\( -\frac{1}{5(5t^{3}+8)} \)
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find the inverse function in slope - intercept form (mx + b): $f(x)=-x …
The inverse function in slope-intercept form is \( f^{-1}(x) = -x - 3 \)
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question given $f(x) = -x^2 - 7x$, find $f(-10)$
\( -30 \)
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problem. 4 : determine the limit. \\(\\lim\\limits_{x\\to -9} \\frac{-5…
\(+\infty\) (or we can say the limit does not exist (DNE) in the set of real numbers, but in the context of limits involving infinity, we can say it approaches \(+\infty\))
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f(x) = -2(x + 3)^2 - 1; f(x) = -2(x + 3)^2 + 1; f(x) = 2(x + 3)^2 + 1; …
Top graph: $f(x)=2(x+3)^2+1$ Middle graph: $f(x)=-2(x+3)^2+1$ Bottom graph: $f(x)=2(x-3)^2+1$
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crusade. after only seven weeks under siege, acre fell, effectively end…
1. Pope Urban II called the First Crusade in 1095 to respond to Byzantine requests for aid against Seljuk Turks, unify Western Christendom, assert papal authority, and reclaim Jer…
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write the following expression in simplest radical form.\\(\\left(\\sqr…
\( 5z\sqrt[3]{25z^2} \)
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the polynomial function $f(x)$ is graphed below. fill in the form below…
The degree of \( f(x) \) is \( 4 \) and the leading coefficient is \( \text{positive} \). There are \( 2 \) different real zeros and \( 3 \) relative extremes.
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express in simplest radical form given ( x > 0 ).( xsqrt{75x^{3}} + x^{…
\(9x^{2}\sqrt{3x}\)
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select the correct answer. a rotating sprinkler head sprays water as fa…
C. $\frac{800}{9}\pi \text{ ft}^2$
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given $f(x) = x^2 - 3x + k$, and the remainder when $f(x)$ is divided b…
\( k=-1 \)
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worksheet a: (topic 2.14) logarithmic modeling name: 1. selected values…
The two equations are \( 2=a + b\ln 1 \) (or \( 2 = a \)) and \( 10=2 + b\ln 6 \) ### Part (b)
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consider the following equation. \\log_{9} 5^{x + 2} = 16 find the valu…
\( x\approx19.843 \)
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topic 5: arithmetic and geometric sequences 25) determine if the sequen…
25) Arithmetic sequence; $a(n)=60-10n$ 26) Geometric sequence; $g(n)=50(0.5)^{n-1}$ 27) 42, 45, 48, 51 28) 12, 36, 108, 324 29) $-376$ 30) No integer term equals 196; $n=38.6$ 31)…
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what are some common sources of bias in ai systems? training data, mode…
A. Training data, model creator, guardrails, user’s prompt
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graph the following equation: y = 2x
A straight line passing through the points $(-1,-2)$, $(0,0)$, and $(1,2)$, extending infinitely in both directions along the grid.
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which is the graph of $f(x) = 100(0.7)^x$?
The top graph with points $(0,100)$ and $(2,30)$
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which graph represents exponential decay?
The top graph (the first graph shown, where $y$ decreases as $x$ increases starting from a high $y$-value) represents exponential decay.
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y = \\frac{2}{3}x + 6 click to select points on the graph.
Key points to plot: \( (0, 6) \) and \( (3, 8) \) (or others like \( (-9, 0) \)), then draw the line.
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given the function $h(x) = x^2 + 3x - 1$, determine the average rate of…
$1$
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what is the slope of this line? simplify your answer and write it as a …
# Explanation: ## Step1: Recall the slope formula The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). ## Step2:…
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solve the following system of equations graphically on the set of axes …
The solution to the system of equations is \( (4,3) \) (the point of intersection of the two lines \( y = 2x-5 \) and \( x + 2y = 10 \)).
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select the correct answer. in which direction must the graph of f(x) = …
B. down
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the function f is defined as follows. $f(x) = -2x^2 + 7$ if the graph o…
$g(x) = -2x^2 + 10$
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find the product. \\(\frac{3}{4} \\times 12 = \\square\\)
9
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type the correct answer in the box. use numbers instead of words. what …
2722
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draw a line representing the ise\ and a line representing the un\ of th…
The slope of the line is \(3\).
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graph this function: $y = x - 9$ click to select points on the graph.
Plot the points $(0, -9)$, $(9, 0)$, and $(5, -4)$, then draw a straight line through them to graph $y=x-9$.
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select the correct answer from each drop - down menu. complete the stat…
1. The vertex of the function is a **maximum** 2. On the interval $(-1, \infty)$, the function is **decreasing** 3. As $x$ approaches positive infinity, the function approaches **…
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the function f is defined as follows. $f(x)=\\sqrt3{x}+2$ find $f(64)$ …
\( f(64) = 6 \) \( f(-216) = -4 \)
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graph the equation. $y = 4|x - 4|$
The graph is a V-shaped absolute value function with vertex at $(4, 0)$, passing through $(3, 4)$ and $(5, 4)$, opening upwards with a slope of $4$ for $x>4$ and $-4$ for $x<4$.
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graph the equation shown below by transforming the given graph of the p…
The graph of $y = x^2 + 5$ is the parent parabola $y = x^2$ shifted vertically upward by 5 units, with key transformed points $(0,5)$, $(\pm1,6)$, $(\pm2,9)$, $(\pm3,14)$ connecte…
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the function $y = f(x)$ is graphed below. what is the average rate of c…
$-4$