Math Quadratic Function Question?

Question

Determine the equation of the line that passes through the points of intersection of the graphs of quadratic functions f(x)=x^2-4 and g(x)= -3x^2+2x+8

Answer 1

f(x)=x^2-4 and g(x)= -3x^2+2x+8

We want where they intersect so set them equal to each other:

x^2-4=-3x^2+2x+8

and solve for x:

4x^2-2x-12=0, 2x^2-x-6=0

Using the quadratic equation we get:

X=2, and -3/2

To find the y coordinate, simply plug in the x values into one of the equations:

x^2-4: such that y is 0 and -7/4, and two ordered points: (2, 0) and (-3/2, -7/4).  

Now we want the equation of the line, so find the slope of the two lines:

(0--7/4)/(2--3/2)=(7/4)/(7/4)=1, so the slope is 1

(y2-y1)=m(x2-x1) and plug in a point and the slope:

y=x-2 is the equation of the line that passes through the intersection of the two points of the graph.  

Answer 2

x^2=4;

x=2,-2;

g(2)=24;

g(-2)=16;

Eqn of line going through(2,24);(-2,16)

(y-24)/(24-16)=(x-2)/(2+2)

y-24=2x-4

y=2x+20;

Answer 3

Solve them simultaneously then you get 4 values of x . then find eqn of line

Answer 4

Set the two equations equal to each other and solve for the intersection.

x^2-4 = -3x^2 + 2x +8

4x^2 - 2x -12 = 0

2x^2 - x - 6 = 0

(2x +3)(x - 2) = 0

then x = -3/2 or x = 2

The points are (2, 0) and (-3/2, -7/4)

So the line y = (-7/4)/(-3/2 -2) (x - 2)

                

finally, y = 1/2(x-2)

which is the same as y = x/2 -1

Answer 5

The points of intersection are found by solving

-3x^2 + 2x + 8 = x^2 - 4

-4x^2 + 2x + 12 = 0

2x^2 - x - 6 = 0

x = [1 +/- √( 1 - 4(2)(-6) ) ] / 2(2)

x = [ 1 +/- √( 49) ] / 4

x = -3/2 and x = 2

f(-3/2) = g(-3/2) = 9/4 - 4 = (9 - 16)/4 = -7/4

f(2) = g(2) = 4 - 4 = 0

The two points are (-3/2,-7/4) and (2,0)

Using the slope-intercept form and the point (2,0), you have

y = mx + b

0 = 2m + b

Using the slope-intercept form and the point (-3/2,-7/4) you have

-7/4 = -3m/2 + b

-7 = -6m + 4b

The system is

2m + b = 0

-6m +4b = -7

The solution is m = 1/2 and b = -1

So the equation is

y = (1/2)x - 1

Now check with a graphing calculator...