Question 1

Graph the quadratic function. 
 $f ( x ) = - \frac { 1 } { 3 } x ^ { 2 } - 2$  
A)   
B)   
C)   
D)

Accepted Answer

The answer of Graph the quadratic function. 
 \(f (...

Question 2

Graph the quadratic function. F (x) = x² + 2x A) B) C) D)

Accepted Answer

The answer of Graph the quadratic function. ​
F (x) =...

Question 3

Find two positive real numbers whose product is a maximum and whose sum of the first number and four times the second is 160.
A) $128,8$
B) $88,18$
C) $80,20$
D) $96,16$
E) $56,26$

Accepted Answer

Given the sum of the first number $x$ and four times the second number $y$ is 160, we can write this as $x + 4y = 160$. To maximize the product $xy$, we can use the method of Lagrange multipliers or simply solve for one variable in terms of the other and substitute into the product equation. Solving for $x$ gives $x = 160 - 4y$. The product $xy$ becomes $(160 - 4y)y$. Differentiating this product with respect to $y$ and setting the derivative equal to zero will give the values of $y$ that maximize or minimize the product. Alternatively, recognizing this as a quadratic equation in terms of $y$, we know that the product is maximized when $y$ is at the vertex of the parabola, which occurs at $y = -\frac{b}{2a}$ when the product is in the form $ay^2 + by + c$. Substituting $x = 160 - 4y$ into $xy$ gives a quadratic in $y$, and finding the vertex gives the optimal $y$. Solving for $y$ and then $x$ gives the pair $x = 80, y = 20$, which corresponds to choice C.

Question 4

Find the vertex of the parabola. y = - x² + 16x - 60 A)(-8, 4) B)(8, 4) C)(-8, -4) D)(8, -30) E)(4, 8)

Accepted Answer

The vertex form of a parabola is $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex. To find the vertex of the given parabola $y = -x^2 + 16x - 60$, we complete the square. The equation can be rewritten in vertex form by completing the square on the $x$ terms:1. Factor out the coefficient of $x^2$, which is -1: $y = -(x^2 - 16x) - 60$.2. To complete the square, add and subtract $\left(\frac{16}{2}\right)^2 = 64$ inside the parenthesis: $y = -(x^2 - 16x + 64) - 60 + 64$.3. Simplify: $y = -(x - 8)^2 + 4$.Thus, the vertex is $(8, 4)$.

Question 5

The function $f ( x ) = x ^ { 2 } - 5$ is one-to-one on the domain $( x \leq 0 )$ .Find $f ^ { - 1 } ( x )$ .

A) $f ^ { - 1 } ( x ) = x ^ { 2 } + 5$
B) $f ^ { - 1 } ( x ) = \sqrt { x + 5 }$
C) $f ^ { - 1 } ( x ) = - \sqrt { x + 5 }$
D) $f ^ { - 1 } ( x ) = \frac { 1 } { x ^ { 2 } - 5 }$
E) $f ^ { - 1 } ( x ) = \sqrt { x - 5 }$

Accepted Answer

To find the inverse function $f^{-1}(x)$, we swap $x$ and $y$ in the original function and solve for $y$. Starting with $y = x^2 - 5$, we swap to get $x = y^2 - 5$, and then solve for $y$, giving $y = \sqrt{x + 5}$. However, since the function is defined as one-to-one on the domain $x \leq 0$, the correct inverse must produce negative outputs for the given domain, making the correct answer $y = -\sqrt{x + 5}$.

Question 6

Find the vertex of the parabola. y = x² + 14x + 49 Please enter your answer as an ordered pair.

Accepted Answer

The answer of Find the vertex of the parabola.
​
y =...

Question 7

Graph the given function. 
 $f ( x ) = 2 -$ $x ^ { 2 }$  
A)  
B)   
C)   
D)   
E)

Accepted Answer

The answer of Graph the given function. 
 \(f (...

Question 8

Find the inverse of the one-to-one function. 
 $y = 7 x + 5$  
A) $y = \frac { x - 7 } { 5 }$
B) $y = \frac { x - 5 } { 7 }$
C) $y = \frac { 7 } { x - 5 }$
D) $y = \frac { x + 5 } { 7 }$
E)none of the above

Accepted Answer

To find the inverse, swap $x$ and $y$ and solve for $y$. So, $x = 7y + 5$ becomes $y = \frac{x - 5}{7}$.

Question 9

Find the vertex of the parabola. Y = 3x² - 1 A)(-1, 3) B)(0, -1) C)(3, -1) D)(-1, 0) E)(0, 3)

Accepted Answer

The given equation is in the form $y = ax^2 + bx + c$, where $a = 3$, $b = 0$, and $c = -1$. The vertex of a parabola in this form is given by $(-b/2a, c)$. Substituting $a = 3$ and $b = 0$ into the formula gives the vertex as $(0, -1)$.

Question 10

A Norman window has the shape of a rectangle surmounted by a semicircle as in the figure below.If the perimeter of the window is 40 ft, express the area, A, as a function of the width, x, of the window. 
A) $A ( x ) = \frac { 160 x - 4 x ^ { 2 } - \pi x ^ { 2 } } { 8 }$
B) $A ( x ) = \frac { 80 x - x ^ { 2 } } { 2 }$
C) $A ( x ) = \frac { 80 x - 2 x ^ { 2 } - \pi x ^ { 2 } } { 4 }$
D) $A ( x ) = \frac { 80 x + ( 2 - \pi ) x ^ { 2 } } { 4 }$
E) $A ( x ) = \frac { 160 x - 4 x ^ { 2 } - \pi x ^ { 2 } } { 2 }$

Accepted Answer

The answer of A Norman window has the shape of...

Question 11

Find the vertex of the parabolic graph of the equation. y = 4(x - 5) ² + 7 A)(5, 0) B)(25, 7) C)(5, 4) D)(7, 5) E)(5, 7)

Accepted Answer

The vertex form of a parabola is $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex. The given equation is in the form of $y = 4(x - 5)^2 + 7$, making the vertex $(5, 7)$.

Question 12

A farmer has 216 feet of fencing and wants to build two identical pens for his prize-winning pigs.The pens will be arranged as shown.Determine the dimensions of a pen that will maximize its area. 
A) $7 ^ { \circ } \times 176$
B) $21 ^ { \prime } \times 52 ^ { 2 }$
C) $27 ^ { \circ } \times 72 ^ { 2 }$
D) $9 \times 108$
E) $27 ^ { \circ } \times 36 ^ { 6 }$

Accepted Answer

The answer of A farmer has 216 feet of fencing...

Question 13

Graph the quadratic function. F (x) = - 4x² + 2 A) B) C) D)

Accepted Answer

The answer of Graph the quadratic function. ​
F (x) =...

Question 14

The height, h(x), of a punted rugby ball is given by $h ( x ) = - \frac { 1 } { 64 } x ^ { 2 } + \frac { 21 } { 32 } x + 3$ where x is the horizontal distance in feet from the point where the ball is punted.How far, horizontally, is the ball from the kicker when it is at its highest point?
A) $45 \text { feet }$
B) $21 \text { feet }$
C) $26 \text { feet }$
D) $31 \text { feet }$
E) $24 \text { feet }$

Accepted Answer

The answer of The height, h(x), of a punted rugby...

Question 15

Find the vertex of the parabola. y = x² - 12x + 36 A)(6, 0) B)(0, 6) C)(-24, 0) D)(36, 0) E)(6, -24)

Accepted Answer

Firstly, rewrite the equation in vertex form: y = (x-6)^2 - 12. We can see that the vertex is at (6,-12/1) which simplifies to (6,0). Therefore, the correct answer is A)(6,0).

Question 16

Graph the quadratic function. F (x) = - x² - 4x + 1 A) B) C) D)

Accepted Answer

The answer of Graph the quadratic function. ​
F (x) =...

Quiz 2: Polynomial and Rational Functions

Graph the quadratic function. $f ( x ) = - \frac { 1 } { 3 } x ^ { 2 } - 2$

Graph the quadratic function. F (x) = x² + 2x

Find two positive real numbers whose product is a maximum and whose sum of the first number and four times the second is 160.

Find the vertex of the parabola. y = - x² + 16x - 60

The function $f ( x ) = x ^ { 2 } - 5$ is one-to-one on the domain $( x \leq 0 )$ .Find $f ^ { - 1 } ( x )$ .

Find the vertex of the parabola. y = x² + 14x + 49 Please enter your answer as an ordered pair.

Graph the given function. $f ( x ) = 2 -$ $x ^ { 2 }$ $Graph the given function. f ( x ) = 2 - x ^ { 2 } A) B) C) D) E)$

Find the inverse of the one-to-one function. $y = 7 x + 5$

Find the vertex of the parabola. Y = 3x² - 1

Find the vertex of the parabolic graph of the equation. y = 4(x - 5) ² + 7

Graph the quadratic function. F (x) = - 4x² + 2

The height, h(x) , of a punted rugby ball is given by $h ( x ) = - \frac { 1 } { 64 } x ^ { 2 } + \frac { 21 } { 32 } x + 3$ where x is the horizontal distance in feet from the point where the ball is punted.How far, horizontally, is the ball from the kicker when it is at its highest point?

Find the vertex of the parabola. y = x² - 12x + 36

Graph the quadratic function. F (x) = - x² - 4x + 1

Quiz 2: Polynomial and Rational Functions

Graph the quadratic function. f(x) =−13x2−2f ( x ) = - \frac { 1 } { 3 } x ^ { 2 } - 2f(x) =−31​x2−2

Graph the quadratic function. ​ F (x) = x2 + 2x ​

Find two positive real numbers whose product is a maximum and whose sum of the first number and four times the second is 160.

Find the vertex of the parabola. y = - x2 + 16x - 60

The function f(x) =x2−5f ( x ) = x ^ { 2 } - 5f(x) =x2−5 is one-to-one on the domain (x≤0) ( x \leq 0 ) (x≤0) .Find f−1(x) f ^ { - 1 } ( x ) f−1(x) .

Find the vertex of the parabola. ​ y = x2 + 14x + 49 ​ Please enter your answer as an ordered pair.

Graph the given function. f(x) =2−f ( x ) = 2 -f(x) =2− x2x ^ { 2 }x2

Find the inverse of the one-to-one function. y=7x+5y = 7 x + 5y=7x+5

Find the vertex of the parabola. ​ Y = 3x2 - 1 ​

A Norman window has the shape of a rectangle surmounted by a semicircle as in the figure below.If the perimeter of the window is 40 ft, express the area, A, as a function of the width, x, of the window.

Find the vertex of the parabolic graph of the equation. y = 4(x - 5) 2 + 7

A farmer has 216 feet of fencing and wants to build two identical pens for his prize-winning pigs.The pens will be arranged as shown.Determine the dimensions of a pen that will maximize its area.

Graph the quadratic function. ​ F (x) = - 4x2 + 2 ​

Find the vertex of the parabola. y = x2 - 12x + 36

Graph the quadratic function. ​ F (x) = - x2 - 4x + 1 ​

Graph the quadratic function. $f ( x ) = - \frac { 1 } { 3 } x ^ { 2 } - 2$

Graph the quadratic function. F (x) = x² + 2x

Find the vertex of the parabola. y = - x² + 16x - 60

The function $f ( x ) = x ^ { 2 } - 5$ is one-to-one on the domain $( x \leq 0 )$ .Find $f ^ { - 1 } ( x )$ .

Find the vertex of the parabola. y = x² + 14x + 49 Please enter your answer as an ordered pair.

Graph the given function. $f ( x ) = 2 -$ $x ^ { 2 }$ $Graph the given function. f ( x ) = 2 - x ^ { 2 } A) B) C) D) E)$

Find the inverse of the one-to-one function. $y = 7 x + 5$

Find the vertex of the parabola. Y = 3x² - 1

Find the vertex of the parabolic graph of the equation. y = 4(x - 5) ² + 7

Graph the quadratic function. F (x) = - 4x² + 2

Find the vertex of the parabola. y = x² - 12x + 36

Graph the quadratic function. F (x) = - x² - 4x + 1