Question

Does anyone know the function of these graphs? please help and thank you. ​

Answer 1

Answer:

Step-by-step explanation:

f(x) is an easy one.  Because it's a parabola, it's standard form is

$y=ax^2+bx+c$

But even simpler than that, look at a point on the graph, in particular, (2, 4).  If x = 2 and y = 4, we can square 2 to get 4, so the equation for that is the parent graph, $y=x^2$, plain and simple.

The next one requires a bit of doing.  Pick 3 points on the graph because we have 3 unknowns to find: a, b, and c.  The points that are easy to pick are (0, -2), (2, -4), (-2, -4).  Use the x and y coordinates from each one of those points to fill in the standard form of the parabola.  Because this parabola is "upside down" the leading coefficient is negative.  Start with the first coordinate first:

$(0, -2)-->-2=-a(0)^2+b(0)+c$ which gives us that c = -2.  That's good...one down, 2 to go.

Next we will use the remaining 2 points to create a system of equations that we can solve simultaneously for a and b.  Using the second coordinate pair (2, -4):

$-4=-a(2)^2+b(2)-2$ gives us the simplified equation:

***-2 = -4a + 2b***

I put the stars in front and behind because we will need to come back to that one in a minute.

Using the last coordinate pair (-2, -4):

$-4=-a(-2)^2-b(2)-2$ simplifies down to:

***-2 = -4a - 2b***

Now put these together and solve the system by elimination, and you see that 2b and the -2b cancel each other out, leaving you with -4 = -8a, so a = 1/2.  Now we know a:  1/2 and c:  -2 and we can find b:

If -2 = -4a + 2b, then -2 = -4(1/2) + 2b, and b = 0.  That means that the equation for the upside down parabola is

$y=-\frac{1}{2}x^2-2$