Question

quiz 5
question 1
convert the following to vertex form using the completing the square technique
$f(x)=2x^{2}-4x + 5$
question 2
convert the following to vertex form using the completing the square technique
$g(x)=5(x + 3)(x + 8)$
question 3
what list of transformations correctly describes $h(x)$?
$h(x)=-7(x - 21)^{2}+15$
question 4
using $h(x)=-6(\frac{1}{2}x - 17)^{2}+14$, what would the original point $(-2,4)$ transform into? (first row of table of values)
question 5
what is the width and the length of a rectangle with an area of $60m^{2}$ and has a length that is $7m$ more than the width?
question 6
a torpedo was fired from a flying submarine. what is the height of the torpedo after 9 seconds if the path the torpedo takes is modelled by the following:
$h(t)=-3(t - 8)^{2}+432$
question 7
a vintage watch company is hoping to make a profit. the cost model, in dollars, to produce x watches is $c(x)=320x + 8118$ and their revenue can be modelled by $r(x)=-2x^{2}+600x$
how many watches should be sold to reach maximum profit?
question 8
using the same information and your answer from the previous question, what should the selling price be of each watch to maximize profit?

Explanation:

Question 1

Step1: Factor out the coefficient of $x^{2}$ from the first two terms

$f(x)=2(x^{2}-2x)+5$

Step2: Complete the square inside the parentheses

For $x^{2}-2x$, half of the coefficient of $x$ is $- 1$, and $(-1)^{2}=1$. So we add and subtract 1 inside the parentheses: $f(x)=2(x^{2}-2x + 1-1)+5$.

Step3: Rewrite the expression

$f(x)=2((x - 1)^{2}-1)+5=2(x - 1)^{2}-2 + 5=2(x - 1)^{2}+3$

Step1: Expand the function

$g(x)=5(x^{2}+8x+3x + 24)=5(x^{2}+11x + 24)$

Step2: Complete the square for $x^{2}+11x$

Half of 11 is $\frac{11}{2}$, and $(\frac{11}{2})^{2}=\frac{121}{4}$. So $g(x)=5(x^{2}+11x+\frac{121}{4}-\frac{121}{4}+24)$

Step3: Rewrite the expression

$g(x)=5((x+\frac{11}{2})^{2}-\frac{121}{4}+24)=5((x+\frac{11}{2})^{2}-\frac{121 - 96}{4})=5((x+\frac{11}{2})^{2}-\frac{25}{4})=5(x+\frac{11}{2})^{2}-\frac{125}{4}$

Step1: Analyze the transformations

The parent - function is $y = x^{2}$. The negative sign in front of 7 reflects the graph of $y=x^{2}$ over the $x$-axis. The factor 7 vertically stretches the graph by a factor of 7. The $(x - 21)$ shifts the graph 21 units to the right. The + 15 shifts the graph 15 units up.

Answer:

$f(x)=2(x - 1)^{2}+3$

Question 2