Sabrina is playing a ball with her dog. She throws the ball in a parabolic path that can be modeled by the function y=1/2(x-3)^2+7

Question

Sabrina is playing a ball with her dog. She throws the ball in a parabolic path that can be modeled by the function y=1/2(x-3)^2+7. Her brother, Bobby, is playing in a tree next to her. Bobby shines his laser pointer from the tree in a line that can be modeled by the function y=1/2x+8.5. At what point(s) will the ball and laser beam intersect?

Answer 1

To solve this we need to see where these two functions intersect.

 

Since Sabrina's path is parabolic, modeled by a quadratic, there will be two points of intersection.

 

So solve, we need to set the two equations equal to one another.

 

 

1/2 (x-3)² + 7 = 1/2x + 8.5

 

Let's expand.

 

1/2 (x² - 6x + 9) + 7 = 1/2x + 8.5

 

To solve, let's first multiply both sides by 2

x² - 6x + 9 + 14 = x + 17

 

Let's combine terms

x² - 6x + 23 = x + 17

 

Now we can move all terms to the left side of the equation.

x² - 7x + 6 = 0

 

Fortunately, this is easily factorable.

(x-6) (x-1) = 0

And x = 1,6

 

We need the coordinates of the points where the two functions intersect so we need to find y for each of the x's.

To do this, you can use either function.  Using Bobby's is probably easier.

So,

1/2 (1) + 8.5 = 9

and

1/2 (6) + 8.5 = 11.5

 

And the points of intersection are (1,9) and (6,11.5)

 

If you're not comfortable with factoring, you can use other methods like completing the squares of the quadratic formula. I really like the quadratic formula because it works in any situation.

-b ± √(b²-4ac)

        2a

In this case a = 1, b = -7 and c = 6

-(-7) ± √((-7)-4(1)(6))

             2(1) 

 

7 ± √(49-24)

       2

 

7 ± √25

     2

 

7 ± 5

  2

 

12/6, 2/2

 

6, 1  These are the x values.  You would solve for y the same as above.

 

Hope this helps.

Answer 2

See this graph to see what is being requested:

 

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

 

To find the points of intersection of these two equations the substitution method works best. Since we have two expressions that represent "y" set them equal to each other.

 

1/2(x-3)²+7 = 1/2(x) + 8.5

 

I would multiply every term on both sides of the equation by 2 to eliminate the fraction and the decimal.

 

(x-3)²+14 = x + 17

 

Square the binomial and since this is a quadratic equation set equal to zero and solve by factoring (easiest):

 

x²-6x+9 +14 = x+17

 

x²-7x+6=0

 

(x-1)(x-6) = 0

 

This results in x=1 or x=6.

 

When x = 1, y = 9; when x=6, y = 11.5.

 

Not sure how Sabrina is able to throw the ball in the parabolic both indicated by this function. Usually when a ball is "thrown" the ball is thrown upward, reaches an apex, and then descends. 

Answer 3

In order for the ball and the laser beam to intersect the functions of both the path of the ball and  the line of the laser beam need to be equal to each other.

 

Thus,

 

0.5 ( x^2 - 6x + 9 ) + 7 = 0.5x + 8.5

 

Multiply on both sides by 2  yielding

 

x^2 - 6x + 9 = x + 3

 

x^2 - 7x + 6 = 0

 

(x-6) (x-1) = 0

 

x = 6 & x = 1

 

Now plug in these values of x into either function to find the value of f(x) which is your y

 

y = 10 when x  = 6

 

y =  9  when x  = 1

 

 

Therefore , we have two points at which they intersect (6,10) & ( 1,9) . My calculations were done assuming your

 

function for the laser pointer was x/2 + 8.5 and not 1/ 2x + 8.5. Hope my feedback was useful.

 

Regards,

 

Mayuran