Algebra 2 Systems of Equations 2/26/21

Algebra 2 Systems of Equations 2/26/21

This lesson is interactive. The questions are TIMED and must be answered before the time expires. However, the informational slides are NOT timed. Make sure you are not on a question slide if you need to leave the computer.

READ THIS FIRST

For the following problem, there are 10 slides with questions. You will have 5 minutes per slide to answer, though it should not take you that long for most. Before you continue, make sure you have sufficient time to answer the questions before the timer runs out (maximum 50 minutes). The timer is the bar running across the top of the screen during the timed slides.

Unanswered questions will be marked as incorrect. These are MAXIMUM times, so you may complete them in less time, depending on your skill level.

Multiple Choice

A rocket is launched from the ground and follows a parabolic path represented by

 $f\left(x\right)=-x^2+10x$    Afterwards, a laser beam is shot from a height of 10 feet and follows a straight path represented by 
 $g\left(x\right)=-x+10$  
We can see their two intersections in the graph.  Let's back that up with algebra.  Since we are looking for locations where f(x) = g(x), we can replace f(x) with  $-x^2+10x$  and replace g(x) with  $-x+10$    What is the resulting equation?

•  $0=-x^2+10x-x+10$  

•  $-x^2+10x\ =\left(-x+10\right)^2$  

•  $-x^2+10x\ =-x+10$  

•  $-x^2+10x\ =\ -x^2+10x$  

Multiple Choice

What is the next step in solving this equation?

• Move all terms to one side of the equal sign and zero on the other.

• Factor the quadratic side (the side with x squared)

• Subtract the x from the $x^2$

• Isolate the $x^2$ and then square root both sides.

Multiple Choice

When solving quadratics, it is recommended that you start with a positive x squared term. Therefore, let's move all our terms to the right side of equal. What is the resulting equation?

• $0=x^2-10x+10$

• $0=x^2-20x$

• $0=-10x+10$

• $0=x^2-11x+10$

Multiple Choice

Fortunately for us, this quadratic trinomial is easily factorable since the leading coefficient is one.  We just need to find two terms that add to be -11x and ____.

• are both positive

• have a product of  $10x^2$  

• subtract to give us -9x

• divide to give us -11x

Multiple Choice

What two terms add to be -11x and at the same time have a product of  $10x^2$  ?

• -10x and -1x

• -11x and 0x

• -10x and 9x

• 10x and 1x

Multiple Choice

What does our equation look like after we factor the trinomial using the -10x and the -1x?

•  $0=\left(x-10\right)\left(x+1\right)$ 

•  $0=\left(x+10\right)\left(x-1\right)$ 

•  $0=\left(x+10\right)\left(x+1\right)$ 

•  $0=\left(x-10\right)\left(x-1\right)$ 

Multiple Choice

What is our next step to find the value(s) for x?

• Multiply the binomials together using FOIL or the box method.

• Set each binomial equal to zero and solve for x.

• Replace the first x with -10 and the second x with -1

• Replace x with zero in both binomials.

Multiple Choice

Solve each binomial for x. What are the two solutions?

• x = 10 and x = -1

• x = 10 and x = 1

• x = -10 and x = -1

• x = 0 and x = 0

Multiple Choice

Now that we know that x is 10 and x is 1, what do we do next?

• Choose one of the two equations, replace x with 10 and solve for y. Then replace x with 1 and solve for y. The two resulting points are the two intersection points.

• Choose one of the two equations and replace y with 10, solve for x, then replace y with 1 and solve for x. The two resulting points are the two intersection points.

• There's nothing left to do. The intersection of the two functions is (10, 1)

• Choose one equation and replace the x with 10 and the y with 1. Then solve it.

Multiple Choice

Choose one of the two equations (I suggest using the linear equation), replace x with 10 and solve for y, note that g(x) is the same thing as y and f(x) is the same thing as y.

Next, take that same equation and replace x with 1 and solve for y.

Where are the two locations where the laser crosses the path of the rocket? Click on the image to enlarge it and click the white x to return.

• (0,10) and (9, 1)

• (10, -1) and (0,0)

• (10, 0) and (-1, 0)

• (10,0) and (1,9)

If you wish, take a break.

When you are ready to start the next problem, click the arrow button below.

Problem 2

There are 10 slides with a maximum time limit of 5 minutes each (50 minutes maximum total time). For most slides, you should be able to answer in less than 5 minutes. Click Next when you are ready to begin.

Multiple Choice

A seabird in the air drops his crab from a height of 30 feet. The distance the crab is from the water can be described with the equation  $h\left(t\right)=-16t^2+30$  , where h is the height (feet) and t is the time (seconds).  Another seabird sees this and flies towards the crab using the path  $g\left(t\right)=-8t+15$  . Can the second bird catch the dropped crab before it hits the water?

Using desmos, graph both of these equations and choose one of the answers below:

• Yes, the second bird will catch the crab before it hits the water

• No, the second bird will not catch the crab before it hits the water.

• The graphs are inconclusive.

Multiple Choice

It appears it's not the crab's lucky day after all.

We know Desmos will tell us those intersection points but we are going to work them out algebraically.

Based on what you learned in the previous problem, what should be our first step?

• add g(t) to f(t)

• subtract f(t) - g(t)

• Set g(t) equal to f(t)

• find

  $f\left(g\left(t\right)\right)$

Multiple Choice

We need to find the points where the functions are both the same. Therefore, we set g(t) = f(t). Replace g(t) and f(t) with their functions and determine which one below is the correct result:

(Click the image to enlarge, click the white x to come back)

•  $\left(-8t+15\right)\left(-16t^2+30\right)=0$ 

•  $-8t+15+-16t^2+30=0$ 

•  $-8t+15=-16t^2+30$ 

•  $-8t+15=16t^2-30$ 

Multiple Choice

To solve this by hand we need to move all the terms to one side so zero will be on the other. Let's keep our squared term positive. What is the resulting equation?

• $16t^2-8t+45=0$
  

• $8t^2-15=0$
  

• $8t^2+15t-30=0$
  

• $16t^2-8t-15=0$
  

Multiple Choice

We COULD factor this quadratic trinomial. We would have to find factors of  $\left(16t^2\right)\left(-15\right)$   that add to be -8t. However, this time let's use the quadratic equation  $t=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$  

Identify the values for a, b, and c
(click the image to enlarge it, click the white x to return)

• a = -16; b = 8; c = -15

• a = 16; b = -8; c = -15

• a = -8  b = 16; c = -15

• a = 16; b = -15; c = -8

Multiple Choice

 $t=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$  
In the quadratic formula, the discriminant is the name of the part under the square root:  $\sqrt{b^2-4ac}$  
It is important because if this value turns out to be positive, we have 2 real solutions.  If it's negative we have 2 imaginary solutions.  If it's zero, we have one real solution.
Replace b with -8, a with 16, and c with -15.  What is the value of the discriminant?

•  $\sqrt{1024}$  

•  $\sqrt{1640}$  

•  $\sqrt{0}$  

•  $\sqrt{-896}$  

Multiple Choice

Since the discriminant is  $\sqrt{1024}$  =  $\pm$  32 , this means we have:

• One real solution for t

• Two real solutions for t

• Two imaginary solutions for t

• No solution for t

Multiple Choice

Replacing the discriminant with its value of 32, we get this equation, which is really two equations (one using plus and the other using minus). Simplify this. What are the two values for t?

• t = 0 and t = 1.5

• t = 9 and t = 7

• t = -1.25 and t = 0.75

• t = 1.25 and t = -0.75

Multiple Choice

Using the two values for t (which, if you recall represents time). We can use either equation but let's use the linear one as it's easier. Find the value of g(t) for each value of t.
When t = 1.25, g(t) = ___ and when t = -0.75, g(t) = ____.
In case it's too small to read, here's g(t):  $g\left(t\right)=-8t+15$  

• g(1.25) = 5
  g(-0.75) = 21

• g(1.25) = 21
  g(-0.75) = 5

• g(1.25)=25
  g(-0.75) = 9

• g(1.25) = 9
  g(-0.75) = 25

Multiple Choice

So we have our two intersection points (1.25, 5) and (-0.75, 21). Which one of the following is the best interpretation of these points in the context of our bird/crab story.

• It took the first bird 1.25 seconds to drop the crab. Later in time, the second bird caught the crab just above the water at 0.75 feet.

• 1.25 seconds after bird 1 dropped the crab, the second bird caught it 5 feet above the water. The point (-0.75, 21) is not relevant to our story because it happened before the crab was dropped.

• 0.75 seconds after the first bird dropped the crab, the second bird caught it 21 feet above the water. 1.25 seconds later, the the second bird crashed into the ocean, along with his crab dinner.

• 5 seconds after the first bird dropped the crab, the second bird caught it just 1.25 feet above the ocean.

Open Ended

Make up a 3 digit number and type it below. This will be your code in case there's any question about this being your work. Remember it or write it down in case I need to ask you for it.

...And you're DONE!

Have a fantastic weekend!!!