Assignment 1 due Wednesday, January 10 General notes...

Math 141, Winter 2018 Instructor – Linda C. Stephenson Assignment 1 – Solving Equations

of 12

Assignment 1 – due Wednesday, January 10

Assignment Description:

The purpose of this assignment is to use a plotting applet to plot and solve

various types of functions. We will use this particular applet (or a similar one)

frequently. It is found at the following website:

Desmos: https://www.desmos.com/calculator

See the next page for basic instructions on using the applet.

General notes on assignments:

Always read and follow the instructions for assignments very carefully.

I will give very specific instructions, and I expect you to follow them!

Also, make sure that you read the “What you Need to Turn In” section

very carefully, to make sure that your assignment is complete.

We will use some applets in multiple assignments, and I will only give a

detailed explanation once, so hold on to your assignments for future

use.

I am ALWAYS happy to “pre-check” your assignments before you turn

them in. Just bring them to class before they are due (NOT on the due

date), and I will look them over. The whole point is to LEARN from the

assignments, and if they are all wrong, that won’t be accomplished.

Read the separate file (posted) about ‘Printing Out Applet Output’ – it

describes the preferred method for printing out graphs and stuff.

https://www.desmos.com/calculator


of 12

The main page looks like this:

Notice that Desmos only recognizes the variable x, so enter all functions in terms

of x.

Introduction to Using Desmos:

Here is an example demonstrating how to use Desmos for this assignment. You

may want to do this example on the computer as you are reading it, so that you

can follow along and learn how to use the applet.

In this assignment, you will use the applet to plot and solve various types of

equations. Basically, you want to use the graph to find the point (or points) of

intersection of two functions.


of 12

Example: 15 xx

This is an example of a radical equation. The solutions to the equation are all

values of x which make the equation true, or which make the left-hand side equal

to the right-hand side. Therefore, the goal is to find these values of x.

We can visualize this graphically by creating a “system” of equations, where one

equation consists of the left-hand side and the second equation consists of the

right-hand side, or:

(1) y = x (left-hand side)

(2) y = 15 x (right-hand side)

Now you want to plot the two functions to see where they intersect (we did an

example of this in class the other night).

To plot more than one function at once on Desmos, enter each function in the

boxes labeled ‘1’, ‘2’, etc. in the upper left-hand corner. CHECK and make sure

that what you typed in looks correct!


of 12

The solution to the original equation is the point of intersection, which I have

highlighted, or the point (4, 4). In other words the solution to the equation is x =

4, because this satisfies the equation: it makes the left-hand side equal to the

right-hand side.

Now, let’s check the solution by plugging that value of x back into the original

equation:

15 xx

When x = 4: 1544 True!

In this case, the “solution” that we quote is just the x-value, not the entire point,

so the answer would be: x = 4. Note that it is possible to have more than one

solution (as you will see), unlike a system of two different lines which can only

intersect in a single point.

Also, you can solve the problem analytically, or in other words, by hand. Use the

method we went over in class:

Isolate the radical on one side: 𝒙 − 𝟏 = √𝒙 + 𝟓

Square both sides: (𝒙 − 𝟏)𝟐 = (√𝒙 + 𝟓)𝟐

x2 – 2x + 1 = x + 5

x2 – 3x – 4 = 0

(x – 4)(x + 1) = 0

x = 4 or x = –1 are the possible solutions.

Check: x = 4: 1544 True!

x = –1: 1511 NOT! Not a solution!

Therefore, the analytic solution confirms that x = 4 is the only solution.


of 12

KEY POINT! The WHOLE point of doing this is to use the graph to confirm

your analytic solution. They should be the SAME!!! If your graph and

analytic solution are not the same, then one of them is wrong, and you

need to check your work! Don’t forget this!!!

HOW TO PRINT YOUR GRAPHS:

There are various methods for doing this, and I will let you guys decide which

method you prefer. We looked at this briefly in class the other night. In the

upper right corner, go to the little arrow icon which leads you to options to ‘Share

you graph’. You can try any of those options: print, export, or embed, and see

what works best. I chose to not use ANY of those, because I am a huge fan of

the Windows ‘Snipping’ tool. There’s a discussion on how to use that in a file

posted on our Assignments page, ‘Printing out Applet Output’. I can do a demo

in class also.

WHAT YOU NEED TO TURN IN:

Fully completed Answer Sheets (5 pages total).

Print out of one plot for each problem (5 plots total), with all points of

intersection (solutions) circled on the plots (problems 1-4).

When you snip and copy the plots (or otherwise print), make sure you get

the ENTIRE output window, including the input functions. (If there’s

something wrong with your graph, and I can’t see the functions you typed

in, then I can’t tell what went wrong).

Label all plots with the problem number, and attach everything in order!


of 12

Problems:

1. Solve the radical equation: 1495 xx

syntax to enter in the function box:

left-hand side: sqrt(5x-9)-sqrt(x+4) OR use sq root tool right-hand side: y=1 (note: this will just plot a horizontal line)

Note: use the right-arrow key to get out from “under” the radical symbol. Print out your Desmos plot showing the graphical solution. There is only

one solution to this one, and it is a nice integer value of x. Circle the point of intersection by hand on the graph.

2. Solve the equation: (x2 + 2x)2 – 8(x2 + 2x) + 15 = 0 syntax:

left-hand side: (x^2+2x)^2-8(x^2+2x)+15 right-hand side: y=0 (note: this will just plot a horizontal line)

There are four solutions to this one, two of which are nice integer values of

x. You can click on the points of intersection with your mouse, and a dot will appear, along with the coordinates of the point. This will give you an estimate of the solution (the x-value) to the nearest thousandth. Print out your plot showing the graphical solution, and circle the points of intersection by hand on the graph.

3. Solve the rational equation: 𝒙+𝟒

𝒙+ 𝟏 =

𝒙−𝟔

𝒙

syntax:

left-hand side: (x+4)/x+1 Note: hit right-arrow key after doing the (x+4)/x

right-hand side: (x-6)/x Print out your plot showing the graphical solution, and circle the point of

intersection by hand on the graph.

For example, I clicked on the point of intersection of these two graphs. The x-coordinate of the “red dot” point are displayed to the right. The graphs intersect when x is about 1.6.


of 12

4. Solve the absolute value equation: |2 – 3x| = 2x – 1 syntax:

left-hand side: abs(2-3x) OR use the |a| tool at the bottom right-hand side: 2x-1

Note: you may want to zoom in on this one in order to see the points of

intersection better. Use the + symbol at the upper right Print out your plot showing the graphical solution, and circle the points of intersection by hand on the graph.

5. A water balloon is catapulted straight up into the air from an initial height of 2.4 meters, such that its height ‘h’ in meters after ‘t’ seconds is given by the following equation:

h = –4.9t2 + 27t + 2.4 Plot the equation with the following syntax, and use the plot to answer all

of the questions: Syntax: -4.9x2+27x+2.4 In particular, notice that the x-axis represents the time ‘t’ in seconds that

the water balloon is in the air, and the y-axis represents the height ‘h’ in meters that the balloon is at, at any particular time. The water balloon is simply traveling straight up and straight down, it is NOT tracing out the path shown in the graph. Make sure you understand what the graph is showing.

For this graph, you will need to adjust the y-axis scale. Go into the Graph

Settings by clicking the little wrench at the upper right corner, and change the maximum value for the y-axis to 40.

You can just click around on the graph to read off the coordinates of the points that you need to answer the questions. You can also “grab” the points, and slide them up and down the curve, to get a more accurate answer. Print out your plot showing the graphical solution.

Math 141, Winter 2018 Name: Assignment 1 – Solving Equations

of 12

Answer Sheet

1. Solve the equation: 1495 xx

Solution(s): x =

Remember to attach your plot showing the graphical solution, with the point of intersection circled. Show all work (NEATLY!) here for the analytic solution:

IMPORTANT NOTE: For all problems, you are going to show your work for the analytic (i.e. “by hand”) solution. Therefore, you should list the exact answers from your analytic solutions in the answer spaces given below. Do not list approximate values which may have been obtained from your graphs. Note that your analytic solutions should match your graphical solutions – that is the whole point! So don’t forget to verify that your solutions are correct.


of 12

2. Solve the equation: (x2 + 2x)2 – 8(x2 + 2x) + 15 = 0

Solution(s): x = Remember to attach your plot showing the graphical solution, with the points of intersection circled. Show all work here (NEATLY!) for the analytic solution (using substitution method):


of 12

3. Solve the equation: 𝒙+𝟒

𝒙+ 𝟏 =

𝒙−𝟔

𝒙

Solution(s): x =

Remember to attach your plot showing the graphical solution, with the point of intersection circled. Show all work here (NEATLY!) for the analytic solution:


of 12

4. Solve the equation: |2 – 3x| = 2x – 1

Solution(s): x = Remember to attach your plot showing the graphical solution, with the points of intersection circled. Use the solution and graph to solve the inequality: |2 – 3x| < 2x – 1

List the solution in interval notation: Show all work here (NEATLY!) for the analytic solution (to the equation):


of 12

5. Water balloon problem: No analytic solution is necessary here, just use your graph to answer the following questions. Give all answers to the nearest tenth (one decimal place). How high is the balloon after 1 second? meters For how long is the balloon more than 30 meters high? sec What is the maximum height of the balloon? meters How long is the balloon in the air? sec Remember to attach your plot showing the graphical solution.

Assignment 1 due Wednesday, January 10 General notes...

Documents

Transcript of Assignment 1 due Wednesday, January 10 General notes...