f(x)=2x^2+4x−16. Here's why.
There are lots of things we can do to find which quadratic function is being graphed.
When a function has a positive leading coefficient, both ends when graphed will continue up, and when the leading coefficient is negative, both ends will continue down. Because both ends of the graph are going up, we can rule out f(x) = −4x^2+3x+16
because it has a negative leading coefficient, (-4).
Another simple way to find which function is being graphed is to find the y intercept of each function.
To find the y intercept of a function, substitute 0 into the function.
f(x) = 2x^2+4x−16
f(0) = 2(0)^2+4(0)−16
f(0) = -16.
f(x) = 3x^2−5x+16
f(0) = 3x(0)^2−5(0)+16
f(0) = 16
f(x) = 5x^2−2x−16
f(0) = 5(0)^2 - 2(0) = -16
f(0) = -16.
The y intercept is the point where the function crosses the y axis. As you can see, the y intercept of the graph is -16. Because of this, we can rule out f(x) = 3x^2−5x+16 because its y intercept is positive 16.
Next we can factor each of the two remaining trinomials to determine the x intercepts. I'll explain how to do that if you don't know how.
f(x) = 2x^2+4x−16
First, factor out the Greatest Common Denominator (GCF)
2x^2+4x−16
The GCF is 2, so divide each term by 2.
2x^2+4x−16
2(x^2 + 2x - 8)
Next, multiply the coefficient of the first and last term
1 x -8 = -8
Find a pair of numbers that multiply to give you -8, and add to give you the coefficient of the middle term (2)
Our pair of numbers are -2 and 4, because -2 x 4 = -8 and -2 + 4 = 2.
Replace the middle term with this pair of numbers.
2(x^2 + 4x - 2x - 8)
Now separate the polynomial inside the parenthesis into two groups, and factor out the GCF for each group.
2[x(x + 4) - 2(x + 4)]
As you can see, these two groups have a GCF, which is (x + 4).
Factor that out and you're left with
2[(x - 2)(x + 4)]
Now take each term in parenthesis, set them to 0, and solve for x.
x - 2 = 0
x - 2 + 2 = 0 + 2
x = 2
x + 4 = 0
x + 4 - 4 + 0 - 4
x = -4
The x intercepts are 2 and -4.
Now do the same thing with the other function.
f(x)=5x^2−2x−16
f(x) = 5x^2 + 8x - 10x - 16
f(x) = x(5x + 8) - 2(5x + 8)
f(x) = (5x + 8)(x - 2)
5x + 8 = 0
5x + 8 - 8 = 0 - 8
5x = -8
5x / 5 = -8 / 5
x = -1.6
x - 2 = 0
x - 2 + 2 = 0 + 2
x = 2
f(x)=2x^2+4x−16's x - intercepts are -4 and 2,
f(x)=5x^2−2x−16 's x -intercepts are -1.6 and 2.
As you can see, the graph's x intercepts are -4 and 2.
f(x)=2x^2+4x−16 is the answer.
Hope this helps!!
Let me know if you don't understand anything and I'll try to explain as best I can.
