10. The curve $y = \cos x$ is reflected in the line $y = 1$ and the resulting curve is then translated by $\frac { \pi } { 4 }$ units in the positive $x$-direction. The equation of this new curve is A $y = 2 + \cos \left( x + \frac { \pi } { 4 } \right)$ B $y = 2 + \cos \left( x - \frac { \pi } { 4 } \right)$ C $y = 2 - \cos \left( x + \frac { \pi } { 4 } \right)$ D $\quad y = 2 - \cos \left( x - \frac { \pi } { 4 } \right)$
The graph of $y = \log _ { 10 } x$ is translated in the positive $y$-direction by 2 units. This translation is equivalent to a stretch of factor $k$ parallel to the $x$-axis. What is the value of $k$ ? A 0.01 B $\log _ { 10 } 2$ C $\quad 0.5$ D 2 E $\quad \log _ { 2 } 10$ F 100
The function $\mathrm { f } ( x )$ is defined for all real numbers. Consider the following three conditions, where $a$ is a real constant: I $\quad \mathrm { f } ( a - x ) = \mathrm { f } ( a + x )$ for all real $x$. II $\quad \mathrm { f } ( 2 a - x ) = \mathrm { f } ( x )$ for all real $x$. III $\mathrm { f } ( a - x ) = \mathrm { f } ( x )$ for all real $x$. Which of these conditions is/are necessary and sufficient for the graph of $y = \mathrm { f } ( x )$ to have reflection symmetry in the line $x = a$ ?
When the graph of the function $y = \mathrm { f } ( x )$, defined on the real numbers, is reflected in the $y$-axis and then translated by 2 units in the negative $x$-direction, the result is the graph of the function $y = \mathrm { g } ( x )$. When the graph of the same function $y = \mathrm { f } ( x )$ is translated by 2 units in the negative $x$-direction and then reflected in the $y$-axis, the result is the graph of the function $y = \mathrm { h } ( x )$. Which one of the following conditions on $y = \mathrm { f } ( x )$ is necessary and sufficient for the functions $\mathrm { g } ( x )$ and $\mathrm { h } ( x )$ to be identical?
The following sequence of transformations is applied to the curve $y = 4x^2$ 1. Translation by $\binom{3}{-5}$ 2. Reflection in the $x$-axis 3. Stretch parallel to the $x$-axis with scale factor 2 What is the equation of the resulting curve? A $y = -x^2 + 12x - 31$ B $y = -x^2 + 12x - 41$ C $y = x^2 + 12x + 31$ D $y = x^2 + 12x + 41$ E $y = -16x^2 + 48x - 31$ F $y = -16x^2 + 48x - 41$ G $y = 16x^2 - 48x + 31$ H $y = 16x^2 - 48x + 41$
A sequence of translations is applied to the graph of $y = x ^ { 3 }$ Which of the following graphs could be the result of this sequence of translations? $$\begin{array} { l l }
\text { I } & y = x ^ { 3 } - 3 x ^ { 2 } + 9 x - 27 \\
\text { II } & y = x ^ { 3 } - 9 x ^ { 2 } + 27 x - 3 \\
\text { III } & y = 27 x ^ { 3 } - 9 x ^ { 2 } + x - 3
\end{array}$$
It is given that $f ( x ) = x ^ { 2 } - 6 x$ The curves $y = f ( k x )$ and $y = f ( x - c )$ have the same minimum point, where $k > 0$ and $c > 0$ Which of the following is a correct expression for $k$ in terms of $c$ ?
The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined as $$f(x) = \begin{cases} 2\sin x, & \text{if } \sin x \geq 0 \\ 0, & \text{if } \sin x < 0 \end{cases}$$ Accordingly, which of the following is the image of the open interval $(-\pi, \pi)$ under $f$? A) $[-2,2]$ B) $(-1,2)$ C) $[0,1]$ D) $(0,2)$ E) $[0,2]$
The graph of the function $f ( x ) = x ^ { 2 } - 2 x + 3$ is translated $a$ units to the right and $b$ units downward to obtain the graph of the function $g ( x ) = x ^ { 2 } - 8 x + 14$. Accordingly, what is the value of the expression $| \mathbf { a } | + | \mathbf { b } |$? A) 4 B) 5 C) 6 D) 7 E) 8
The graph of the function $f$ is given below. Given that $\mathbf { g } ( \mathbf { x } ) = \mathbf { 3 } - \mathbf { f } ( \mathbf { x } - \mathbf { 2 } )$, what is the sum $\mathbf { g } ( - \mathbf { 2 } ) + \mathbf { g } ( \mathbf { 5 } )$? A) - 3 B) - 1 C) 1 D) 2 E) 3
Below is the graph of a function $f$. $( a > 2 , b < 1 )$ Accordingly, which of the following could be the graph of the function $| \mathbf { f } ( \mathbf { x } + \mathbf { 2 } ) | - \mathbf { 1 }$? A) [graph A] B) [graph B] C) [graph C] D) [graph D] E) [graph E]
Function f is defined for every $\mathrm { x } \in ( 0,3 ]$ as $$f ( x ) = 2 x + 1$$ and satisfies the equality $$f ( x ) = f ( x + 3 )$$ for every real number x. Accordingly, what is the sum $\mathbf { f } ( \mathbf { 6 } ) + \mathbf { f } ( \mathbf { 7 } ) + \mathbf { f } ( \mathbf { 8 } )$? A) 8 B) 12 C) 15 D) 18 E) 21
In the rectangular coordinate plane, the graphs of functions $\mathrm{f}(\mathrm{x})$ and $\mathrm{g}(\mathrm{x})$ defined on the interval $[0,3]$ are given in the figure. For a number $\mathrm{a} \in (0,1)$, $$\begin{aligned}
& \mathrm{b} = (f \circ g)(a) \\
& c = (g \circ f)(a)
\end{aligned}$$ are determined. Accordingly, which of the following is the correct ordering of the numbers a, b, and c? A) a $<$ b $<$ c B) a $<$ c $<$ b C) b $<$ a $<$ c D) b $<$ c $<$ a E) c $<$ a $<$ b
Let $a$, $b$, and $c$ be real numbers. In the rectangular coordinate plane, the graphs of the functions $f(x) + a$, $b \cdot f(x)$, and $f(c \cdot x)$ are given in the figure. What are the signs of the numbers $a$, $b$, and $c$ respectively? A) $-, +, -$ B) $+, -, +$ C) $-, +, -$ D) $-, -, +$ E) $-, -, -$