2.6 Modelling with quadratics (Pure 1 - Chapter 2: Quadratics)

Summary

This educational video script covers the application of quadratic functions in real-world scenarios, specifically modeling with quadratics. It explains how to solve quadratic equations using factorization, quadratic formula, and completing the square. The script uses a scenario where a spear is thrown from a tower, and the height of the spear is modeled by a quadratic function. It guides through interpreting the constant term, finding when the spear hits the ground, rewriting the function in vertex form, and determining the maximum height reached by the spear. The lesson emphasizes the importance of understanding the context and applying mathematical skills to solve practical problems.

Takeaways

• 📚 The section focuses on modeling with quadratics, which is a common application in higher-level math.
• 🔍 Quadratics can be solved by factorizing, using the quadratic formula, or completing the square.
• 📈 Functions with quadratics are used to model real-world scenarios, such as the trajectory of a thrown spear.
• 📊 When sketching quadratics, it's important to identify the roots, y-intercept, and the maximum or minimum point of the graph.
• 🔢 The discriminant helps determine the nature of the roots of a quadratic equation: positive for two real roots, zero for one repeated root, and negative for no real roots.
• 🎯 In the example, the height of a spear thrown from a tower is modeled by a quadratic function, with the height given in meters after T seconds.
• 🕒 The time it takes for the spear to hit the ground is found by solving the quadratic equation where the height is zero.
• 📉 Completing the square is a method used to rewrite the quadratic equation in a form that makes it easier to identify key features like maximum or minimum points.
• 🏔 The maximum height of the spear above the ground is found by analyzing the completed square form of the quadratic equation.
• ⏱ The time at which the spear reaches its maximum height is determined by setting the expression inside the squared term to zero.

Q & A

What is the main topic of the sixth section of chapter two?

The main topic is modeling with quadratics.

What are the different methods mentioned for solving quadratics?

The methods mentioned are factorizing, using the quadratic formula, and completing the square.

What is the significance of the constant term 12.25 in the model?

The constant term 12.25 represents the initial height from which the spear is thrown, which is 12.25 meters.

How can you determine when the spear hits the ground using the quadratic model?

The spear hits the ground when the height (h(t)) is zero, which means solving the quadratic equation for t when h(t) = 0.

What is the time it takes for the spear to hit the ground according to the model?

The time it takes for the spear to hit the ground is approximately 3.68 seconds.

What is the purpose of completing the square in the context of this quadratic model?

Completing the square helps to rewrite the quadratic function in a form that makes it easier to identify the vertex of the parabola, which can be used to find the maximum or minimum point.

How is the quadratic function h(t) = -4.9t^2 + 14.7t + 12.25 rewritten in the form a - b(t - c)^2?

The function is rewritten as h(t) = -4.9(t - 1.5)^2 + 23.275.

What are the values of a, b, and c in the completed square form of the quadratic function?

In the form a - b(t - c)^2, a is 23.275, b is 4.9, and c is 1.5.

What is the maximum height the spear reaches above the ground according to the model?

The maximum height the spear reaches is 23.275 meters.

At what time does the spear reach its maximum height?

The spear reaches its maximum height at 1.5 seconds, or three halves of a second, after being thrown.

Outlines

00:00

📚 Introduction to Quadratic Modeling

This paragraph introduces the final section of Chapter 2 on quadratics, focusing on modeling with quadratics. It emphasizes the importance of applying various skills learned about quadratics, such as solving them through factorization, using the quadratic formula, or completing the square. The paragraph also touches on the use of functions, sketching, and understanding the roots, y-intercept, and maximum or minimum points of a quadratic graph. The discriminant's role in determining the number of roots is also mentioned. The section begins with an example of a spear thrown from a tower, described by a quadratic function, to illustrate the application of these skills in a real-world scenario.

05:00

🔍 Interpreting the Model and Finding When the Spear Hits the Ground

The speaker interprets the constant term 12.25 in the quadratic model as the initial height from which the spear is thrown. The paragraph then addresses the question of when the spear will hit the ground by setting the height function to zero and solving the resulting quadratic equation. The speaker opts to use a calculator to find the time it takes for the spear to hit the ground, rejecting the negative solution as time cannot be negative. The solution indicates that the spear will hit the ground after approximately 3.68 seconds, rounded to three significant figures.

10:01

📐 Completing the Square to Rewrite the Quadratic Function

The paragraph demonstrates how to rewrite the quadratic function in the form of a completed square, which is a method to make the quadratic equation easier to analyze. The speaker factors out the coefficient of the t^2 term, rearranges the equation, and completes the square to find the values of constants a, b, and c. This process involves creating a perfect square trinomial and adjusting the constant term to maintain equality. The completed square form helps in understanding the vertex of the parabola, which is crucial for further analysis of the spear's trajectory.

15:02

🏔️ Determining the Maximum Height and Time of the Spear's Flight

The final paragraph uses the completed square form to determine the maximum height the spear reaches above the ground and the time at which this maximum height is achieved. The speaker explains that the maximum height occurs when the value inside the squared bracket is zero, leading to the calculation of the time t = 1.5 seconds. The maximum height is found to be 23.275 meters. This section concludes with the practical implication of the model, suggesting that the calculated maximum height and time give people on the ground a chance to take cover from the falling spear.

Keywords

💡Quadratics

Quadratics are mathematical functions of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. They are represented by parabolas and are fundamental in modeling situations where the relationship between variables is not linear. In the video, quadratics are used to model the height of a spear thrown from a tower over time, illustrating how quadratic functions can describe real-world scenarios.

💡Modeling

Modeling in mathematics refers to the process of creating a simplified, often mathematical, representation of a real-world situation or system. It's a way to understand complex phenomena by using mathematical tools. In the context of the video, modeling with quadratics involves using the quadratic function to represent the height of a spear as it travels through the air, demonstrating how mathematical models can predict outcomes.

💡Solving Quadratics

Solving quadratics involves finding the values of the variable that make the quadratic equation true. The video mentions several methods such as factorizing, using the quadratic formula, or completing the square. These methods are crucial for determining the roots of the quadratic equation, which in the video's context, represent the times when the spear hits the ground or reaches its maximum height.

💡Quadratic Equation

The quadratic equation is a standard form of a quadratic function, often written as ax^2 + bx + c = 0. It is used to find the roots of a quadratic function. In the video, the quadratic equation is used to determine when the spear will hit the ground, which corresponds to the height being zero.

💡Completing the Square

Completing the square is a method used to solve quadratic equations by transforming them into a perfect square trinomial plus a constant. This method is demonstrated in the video as a way to rewrite the quadratic function in a form that makes it easier to identify the vertex of the parabola, which represents the maximum or minimum point of the function.

💡Vertex

The vertex of a parabola is its highest or lowest point. In the context of the video, finding the vertex is essential for determining the maximum height the spear reaches. The vertex form of a quadratic function is used to easily identify this point, which is crucial for understanding the behavior of the spear's trajectory.

💡Discriminant

The discriminant of a quadratic equation, denoted as Δ (delta), is a value that determines the nature of the roots of the equation (real and distinct, real and repeated, or complex). It is calculated as b^2 - 4ac. In the video, the discriminant is mentioned as a tool to determine the number of real roots the quadratic equation has, which in this case, relates to the number of times the spear touches the ground.

💡Roots

Roots, also known as zeros or solutions, are the values of the variable that make the quadratic equation equal to zero. They are crucial in the video's context because they represent the times when the spear is at ground level, which is when it is thrown and when it hits the ground.

💡Y-intercept

The y-intercept is the point where the graph of the function intersects the y-axis. It occurs when the input variable (time, in this case) is zero. In the video, the y-intercept is used to determine the initial height from which the spear is thrown, which is a key parameter in the quadratic model.

💡Significant Figures

Significant figures are the digits in a number that carry meaning contributing to its precision. The video emphasizes the importance of rounding answers to an appropriate number of significant figures, which is a common practice in scientific calculations to ensure the answer's precision matches the precision of the original data. In the context of the video, the time it takes for the spear to hit the ground is rounded to three significant figures.

Highlights

The section focuses on modeling with quadratics, emphasizing the application of skills learned about quadratics.

Quadratics can be solved by factorizing, using the quadratic formula, or completing the square.

Functions with quadratics are used to model real-world scenarios, such as the height of a spear thrown from a tower.

The roots of a quadratic equation are important for understanding where the graph intersects the y-axis.

The discriminant helps determine the number of roots a quadratic equation has.

A practical example is given where a spear is thrown from the top of a tower, and its height is modeled by a quadratic function.

The constant term in the quadratic model represents the initial height from which the spear is thrown.

The time it takes for the spear to hit the ground is found by solving the quadratic equation where the height is zero.

Using a calculator is an acceptable method to solve the quadratic equation for the time it takes for the spear to hit the ground.

The solution to the quadratic equation gives the time in seconds when the spear hits the ground.

The quadratic function is rewritten in the form a - b(t - c)^2 to find the maximum height of the spear.

Completing the square is a method used to rewrite the quadratic function in vertex form.

The maximum height of the spear is found by setting the expression inside the brackets to zero.

The time at which the maximum height is reached is determined by solving for t when the bracket expression equals zero.

The completed square form of the quadratic function makes it easier to identify the maximum height and the time it occurs.

The practical application of quadratics in modeling shows the spear reaching its maximum height of 23.275 meters at 1.5 seconds.

Transcripts

this is the Sixth and final section of

chapter two quadratics and this section

is all about modeling with quadratics

so you will come off come across

modeling uh quite a few times

in

um maths in year 12 year 13 and

basically modeling requires us to apply

all the skills we have learned about a

topic

so in this topic with quadratics what

have we learned what we've learned about

solving quadratics and we can solve them

by factorizing by using the quadratic

equation or by completing the square

we also looked at how we use functions

with quadratics

also we looked at sketching quadratics

and when we schedule quadratics we want

to know where the roots are if there are

any

where the graph crosses the y-axis the

y-intercept

and where its maximum or minimum point

is on the quadratic and the last thing

that we looked at was the discriminant

and using the discriminant to work out

how many roots we have if it's positive

two real Roots if it's equal to zero two

repeated Roots basically one root and if

it's less than zero no real Roots so

we'll be using these skills when we're

modeling with quadratics

example 15 a spear is thrown over level

ground from the top of a towel

the height in meters of the spear above

the ground after T seconds is modeled by

the function h of T equals

12.25 plus

14.70 minus at 4.9 t squared where T is

greater than or equal to zero

okay so this is my diagram what's going

on here's the tower somebody stand at

the top of the tower they throw a spear

I imagine the the path of this spear is

going to look something like this it's

going to gradually go down and hit the

ground at some point my first question

would be why is somebody throwing a

spear from the top of a tower very

strange Behavior anyway that's what we

we don't need to answer why they're

fraying a spear we just need to answer

some questions about this model so first

of all in party we need to interpret the

meaning of the constant term 12.25 in

the model so that's this number here

12.25 so when am I just going to get

12.25 well that's going to be when T is

zero when T is zero all of these two

terms here are going to disappear so

I'll just blank them out so we're going

to pretend that they're disappearing and

we're just left with h of T equals 12.25

so let me write that down first so when

T is zero

that means that h of T is 12.25

now the question does say interpret now

interpret is not just writing down a set

of numbers like I've done here it's

looking at the context of the question

so what do we know about when T is zero

well that is when this spear is thrown

yeah

so we want to talk about when the spear

is thrown

and

12.25 is going to be h of T is the

height that's 12.25 meters so basically

we can say this this is going to be our

interpretation this is what we want to

write as an answer or something similar

so the spear

is thrown

from a height

of

12.25 meters there we go there's our

full interpretation that's why we want

to answer it why they're throwing the

spear from the top of a tower who knows

nobody knows very strange Behavior as I

said but that's the height they're

throwing it from and so any any question

that says interpret

then you're going to write some sort of

sentence down look at the context of the

question and

comment using the words that they use in

the questions I'll just highlight the

word interpret okay so let's have a look

at Part B

and Part B

is asking us it says how after how many

seconds does the spear hit the ground

well the spear is going to hit the

ground

and hopefully not a person there's no

one there maybe it's some sort of

Olympic Training or something maybe

that's what Olympians do they throw

Spears from top of towers who knows so

um so it's this the time sorry I'm

written that wrong

I'm concentrating too much on my

person's frame

um this fit the spear so

um

the spear

will

hit the ground hit the ground

when

his height

is zero and that's when h of t

is equal to zero

that's the height so what does that mean

that means that this quadratic that I've

got here is equal to zero now I'm just

going to swap the terms around so I've

got the minus 4.9 t squared first and

then the 14.70 don't have to do this and

then positive plus

2.25 12.25 is next and that's going to

equal to zero so I now need to solve

this quadratic now we've got different

choices to solve the quadratic we could

factorize

we can complete the square so I'll just

put that as CTS complete the square

or we can use the quadratic equation

quadratic equation or we could even use

our calculator

to solve this there's nothing in the

question that says we can't use our

calculator if you can't use your

calculator it'll be like a sentence at

the starting bold that say it says

something like answers which rely wholly

on

um calculated technology will not be

acceptable so this question doesn't say

anything like that

so I'm going to use my calculator and if

you've got the FX

991ex and you're using a calculator you

want to press menu

first once we've pressed the menu button

we want to scroll down until we find our

menu a or num letter A that's for

solving equations

we then press 2 because we want to solve

a polynomial

and we want to solve a polynomial with a

degree of 2 that means the highest power

we use two

you can go right up to a power of four

so that's the key sequence to solve this

on a calculator so that's exactly what

I'm going to do I'm going to use my

calculator to solve this so I'm in the

menu now polynomial degree to and so I

type in negative 4.9 equals

14.7 equals

12.25 equals press equals again and I

get my first solution and it's 3 plus

root 19 over 2 3 plus root 19 over 2.

now I don't really want

um a third answer if it's a time so if I

press the SD button I can see that I get

three point

six seven nine four four and so on

I'll probably want to give that to three

significant figures

so that would be

3.68 seconds

um now in terms of rounding

we normally want to go for three

significant figures as a minimum unless

the question says otherwise round to a

minimum of three significant figures

so minimum

of

free SF unless you are told otherwise

that's what they expect you to do in a

level so don't just do something random

or I feel like writing down everything

on my display or I feel like rounding to

one decimal place always free

significant figures

unless the question says otherwise

so that's an answer let's press equals

to get the next solution and that is 3

minus root 19 over 2. if I press the SD

all right I get a negative answer I'll

get Negative 0.67944

now the model says that t is greater

than or equal to zero so this solution

here we're going to reject it's no good

so we just stick with the first one that

we've written down there's only one

solution to this which makes sense it's

only going to hit the ground once I

guess it's not going to bounce up off

the ground

so 3.68 seconds

give them to free significant figures

part C we're asked to write h of T in

the form a minus B and in Brackets T

minus C all squared where a B and C are

constants to be found now when we write

something in this form

basically I've completed the square it

may look slightly different to how we're

used to seeing it

because normally what we would see is

this part first and then the number at

the end but I'm guessing because it's a

negative they've written the other way

around but it's completing the square

so let's write our quadratic down

minus 4.9 t squared

Plus 14.70

plus 12.25

and what I need to do is to factorize

out the negative 4.9 so we'll do that

first so negative 4.9 and then divide

everything by 4.9

so or negative 4.9 so let's do 14.7

divided by negative 4.9 that's negative

3 so minus 3T

and then we'll do 12.25

divided by negative 4.9 so we're

factorizing that out minus five over two

right so now this is in a format where

we can complete the square

so negative 4.9

and then in Brackets

we'll have T minus

and then half of the three which is

three over two keep as a fraction 3 over

2 squared

now what will this generate what number

will it generate that we don't want

negative or 3 over 2 or squared negative

3 over 2 times negative 3 over Square

two squared I don't want negative 3 over

2 squared

so I'll write it as negative 3 over 2

squared I don't want that I'll take that

off that's the same as 9 over 4 so you

could have written nine over four but I

do want negative 5 over 2. so remember

take off what you don't want which is

this and input in what you do want which

is negative five over two so from here

what I can do

is actually simplify what I've got in

Brackets

so that's going to be T minus three over

two all squared

and then the negative 9 over 4

minus five over two

and that's negative 19 over 4.

now it's not quite in the form that they

want I actually need to

expand the brackets to get exactly in

that form

so that's going to be we'll do over here

negative 4.9

and then T minus three over two all

squared and then I need to do negative

nine over four and times that by

negative 4.9

and that gives 931 over 40. let's see

what that looks like as a decimal oh

that's not too bad 23.275

um because it's a negative times a

negative it's a positive so Plus 23.275

now

um I can see that we've got a B and C so

just for clarity I'll just write down

what a B and C are so my value of a the

number that's on its own is 23

0.275

my value of B

that's the number in front of the

brackets

so it says negative B so I've got

negative 4.9 so B is actually 4.9

and then C

that is the number in the bracket so

that's three over two

um and I'll write that as

um now is that need to be negative or

positive it's already got a negative

there so negative 1.5 I could write

three over two I'm just being consistent

in that everything else is a decimal so

I'll write that as a decimal so a

23.275 b 4.9 and see 1.5

and last part of this question using

your answer to part C or otherwise find

the maximum height of this spear above

the ground and the time at which this

maximum height

is reached so this is using our

completed the square form

and working out what's going on

now since he's talking about a maximum

height I think that the path of my spear

is actually going to be something like

this going up

and then coming down like this and we're

trying to find out

what this is here that's the max height

so my

I've updated my drawing

to what I think actually it's going to

look like we want to work out this

otherwise the maximum height would just

be 12.5 if it was going down but it so

I'm just frame the spear up and then it

comes back down

so let's look back at when we completed

a square so that's this bit here we've

got this negative 4.9 so I'm multiplying

a negative number by something and I

want to work out in a maximum the whole

expression is is going to be

so the maximum this whole expression is

going to be is when I multiply negative

4.9 by the lowest number the smallest

that I can make the total contents of

this bracket

or actually I think using this second

form of the completed the square may be

more useful because a bit clearer to see

what's going on

okay so we're always going to have

23.275 there we can't do anything that's

always going to be there

and then we've got negative 4.9 Times by

something and something squared so we're

going to be multiplying negative 4.9 by

a positive number so always going to get

all of this part here so let me just

highlight just that part so all of this

part here

whatever value of T we have is always

going to end up being a negative number

so we'll always end up by taking

something away from

23.275 unless we can make this part 0

and then we won't be taking anything

away so if we can make this first part 0

then we're not going to be taking

anything away from 23.275

so first of all

the maximum height

the maximum value of H of T is

23.275 meters

that's the maximum value of that whole

expression yeah because I'm not never

going to be adding anything to it I'm

only going to be taking away unless I

make this whole part here zero I won't

be taking anything away and I'll take I

won't be taking anything away if I can

make the brackets 0 and I can make the

bracket 0 when T is three over two so

that maximum height is going to occur

when the time

is three over two seconds or 1.5 seconds

so actually having the completed Square

form

actually in this format

it's easy or easier to see what's going

on in terms of what the whole expression

gives you a positive number taking away

something can we avoid taking anything

away yes when we make t the value 3 over

2 or 1.5 seconds we've got a maximum

height

23.275 meters and that occurs at one and

a half seconds so one and a half seconds

after this person this lunatic has

thrown a spear off of this Tower it's

going to reach its maximum height I

guess it gives some time for the people

that are on the ground to sort of take

cover and run away from this falling

spear

so you should now be able to do exercise

2H

on pages 34 to 35 and then once you've

done that you can think I want to the

mixed exercise