Graphing This System of Equations

Question

The Question: Graph these two equations, then solve for this system of equations by finding the intersection points, rounded to the hundredths place. (Select the correct answer from below.)
y = 2x^2 + 5x – 4
y = -2(x + 3)^2 + 4

 

The Options Given: 

a.) First point: (-0.71,-6.53); second point: (-2.98,4.00)
b.) First point: (0,-4) [exactly]; second point: (-3.54,3.41)
c.) First point: (-0.71,-6.53); second point: (-3.54,3.41)
d.) First point: (0,-14) [exactly]; second point: (-3.54,3.41)
e.) First point: (0,-14) [exactly]; second point: (-2.98,4.00)

 

Thank you!

Answer 1

Well, graphing is nice, but sometimes equating the functions is all you can do.

IF you are graphing, plot both parabolas, then use the intersect feature on your calculator (after adjusting the window such that you can see it) to get the intersection points (that's right, there may be more than one intersection).

 

Analytical solving:

equate the functions:

=> 2x^2 + 5x - 4 = -2(x+3)^2 + 4

Expand the binomial

=> 2x^2 + 5x - 4 = -2(x^2 + 6x + 9) + 4

=> 2x^2 + 5x - 4 = -2x^2 - 12x - 18 + 4

Get the values all on one side:

=> 4x^2 + 17x + 10 = 0

Quadratic formula:

x = [-17 ± √(289 - 160)]/8

*Note, the square root of 129 is close to 11 (so 11.1 is a good approximation)

x = [-17 + 11.1]/8

x = [-17 - 11.1]/8

 

x-values: -0.74, -3.52

So we know the answer is either (a) or (c)

 

y-values (can use either equation, as they both work since it is an intersection of the two)

x = -0.74: y = -2*(-0.74+3)^2 + 4 = -6.2

x = -3.52: y = -2*(-3.52+3)^2 + 4 = 3.46

 

Thus, the points are approximately:

(-0.74,-6.2); (-3.52,3.46)

 

The option closest to our answer is (c)