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Here are a few recommended readings before getting started with this lesson.
Magdalena has just found an old printer toy she used to play with when she was a child. She discovered that the toy prints random functions as labels.
Magdalena ran the printer and got three labels with the following functions.Review the definitions of linear, quadratic, and exponential functions.
Linear Function | $y=2x+1$ |
---|---|
Quadratic Function | $y=x_{2}+3x+1$ |
Exponential Function | $y=3⋅2_{x}$ |
In the following applet, determine whether the given function is linear, quadratic, or exponential.
When considering a table of values, the trend of the data can be determined by observing how the dependent variable changes over equal intervals. To do so, the following situations need to be analyzed.
The applet below illustrates each situation.
Magdalena's father works as a manager for a fish farming company. This morning, Magdalena and her brother Vincenzo poked around in her father's briefcase. They found some papers containing information about the fish population in three nearby lakes.
Fish Population (Thousands) | |||
---|---|---|---|
Time (Months) | Lake Verdastro | Lake Rumoroso | Lake Mezzo |
$0$ | $1356$ | $721$ | $1207$ |
$1$ | $6676$ | $2163$ | $3007$ |
$2$ | $11996$ | $6489$ | $7807$ |
$3$ | $17316$ | $19467$ | $15607$ |
$4$ | $22636$ | $58401$ | $26407$ |
Lake Verdastro: Linear Function
Lake Rumoroso: Exponential Function
Lake Mezzo: Quadratic Function
The $y-$values have a common difference in a linear function, while in an exponential function, the $y-$values have a common ratio. Furthermore, in a quadratic function, the second differences are constant.
Lake | Model |
---|---|
Verdastro | Linear Function |
Rumoroso | Exponential Function |
Mezzo | Quadratic Function |
Note that the analysis suggests that Lake Rumoroso is the best for fish farming. The children gave this information to their father, who was delighted by this excellent analysis and took them for ice cream as a reward.
Magdalena will start college soon, so she decided to start saving money to buy a new computer. The computer she wants costs around $$2000.$ Thanks to her help with the analysis for the fish farming company, her father gave Magdalena an initial amount of $$1000.$ She plans to add $$40$ to her savings each week.
Magdalena wants to buy the computer as soon as possible. She thinks she will be able to afford it after $30$ weeks of saving the money but she isn't sure. Answer the following questions to help her discover if she will save enough money for the new computer by then.
$x=30$
Multiply
Add terms
Magdalena is happy with her new computer. She is also using it for her part-time job as a designer to help pay for her school expenses. She sells $50$ designs per month at $$59$ each. However, because creating each design is complex, she plans to increase the price of each design. She estimates that for each $$5$ increase, two fewer designs will be sold per month.
Magdalena wants to know what price will maximize her revenue, or the amount of money she makes. Identify the following information to help Magdalena make the best decision about her small business.
$S(x)=50−2x$, $N(x)=59+5x$
Distribute $50−2x$
Commutative Property of Multiplication
Distribute $59&5x$
Add terms
Commutative Property of Addition
$a=-10$, $b=132$
$a(-b)=-a⋅b$
$--ba =ba $
Calculate quotient
Vincenzo, Magdalena's little brother, loves dinosaurs. He wants to make a $10-$minute film for a school assignment, explaining the history of this extinguished group of reptiles. He asked Magdalena to help him make and edit the video on her computer. However, the siblings are worried because Vincenzo has a USB drive with only $16GB$ to store the film on.
After some research, Magdalena found that any high-quality video file requires an initial storage space of $20MB.$ This required storage space doubles for each additional minute. The children now have to investigate whether the USB drive has enough space for Vincenzo's completed film. Find the following information to help them.
Function: $f(x)=20⋅2_{x}$
$x=10$
Calculate power
Multiply
See solution.
Begin by analyzing the behavior of the functions. Do all functions have a turning point? What is the maximum number of $x-$intercepts each function could have?
Three distinctive characteristics of quadratic functions will be identified one at a time.
Linear and exponential functions either only increase or only decrease. Conversely, quadratic functions always decrease and increase. The order depends on the direction in which the parabola opens.
Because a parabola either opens upward or downward, there is always one point that is the absolute minimum or absolute maximum of the function. This point is called the vertex.
Using this information, a third distinctive characteristic of quadratic functions can be determined.
While non-horizontal linear functions and exponential functions have at most one $x-$intercept, quadratic functions can have up to two $x-$intercepts.
Only three distinctive characteristics of quadratic functions are presented here. Please note that there are a few others.
Three Distinctive Characteristics of Quadratic Functions |
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Linear and exponential functions either only increase or only decrease. Conversely, quadratic functions always decrease and increase. The order depends on the direction in which the parabola opens. |
Because a parabola either opens upward or downward, there is always one point that is the absolute minimum or absolute maximum of the function. This point is called the vertex. |
While non-horizontal linear functions and exponential functions have at most one $x-$intercept, quadratic functions can have up to two $x-$intercepts. |