Question

Determine the equation of the graph and select the correct answer below. parabolic function going down from the left through the point negative five comma zero and turning at the point negative four comma negative one and going up through the point negative three comma zero and through the point zero comma fifteen and continuing towards infinity Courtesy of Texas Instruments y = (x + 4)2 + 1 y = (x + 4)2 − 1 y = (x − 4)2 + 1 y = (x − 4)2 − 1

Answer 1

1.
A "parabolic" function, is also called a "Quadratic" function.

Any quadratic function can be written in the form:

$f(x)=ax^2+bx+c,$ where a≠0, generally called the "standard form".

The set of all points (x, f(x)) plotted in a coordinate axes, forms a parabola.



2.
The parabola contains the points 

A(-5, 0), B(-4, -1), C(-3, 0), D(0, 15)


from the above discussion we have:


A(-5, 0) = A(-5, f(-5)),

so f(-5)=0

$f(-5)=a(-5)^2+b(-5)+c\\\\0=25a-5b+c$, 


similarly:

$f(-4)=a(-4)^2+b(-4)+c\\\\-1=16a-4b+c$


and 

$f(0)=a(0)^2+b(0)+c\\\\15=c$

the last equation is particularly important, because it tells us that c=15.

3.
using the first 2 equations we write the system of linear equations:


$\begin{cases} i) 25a-5b+15=0\\ii) 16a-4b+15=-1 \end{cases}\\\\\\\\\begin{cases} i) 25a-5b=-15\\ii) 16a-4b=-16 \end{cases}$

divide the first equation by 5, and the second one by -4:

$\begin{cases} i) 5a-b=-3\\ii) -4a+b=4 \end{cases}$

add the 2 equations:

a=1, 
then, substituting in any of the equations:

-4a+b=4
-4+b=4
b=8,

4.

thus the function is $f(x)=x^2+8x+15$


Remark:

given any 3 points of a parabola, it is possible to write the quadratic function.

we did not use 4 of the points, 3 were enough. 

Answer 2

Solution: The correct option is second option, i.e., $y=(x+4)^2-1$.

Explanation:

The standard form of the parabola along the y-axis with vertex (h,k) is $y=a(x-h)^2+k$.

Since the turning point is given as $(-4,-1)$.

Put these values in the standard form of the parabola.

$y=a(x+(-4))^2+(-1)$

$y=a(x+4)^2-1$       .....(1)

The parabola passes through the points (-5,0), (-3,0) and (0,15). It means each point will satisfy the above condition.

Put x = 0 and y = 15 in the equation (1).

$15=a(0+4)^2-1\\16=4^2a\\16=16a\\a=1$

Put a = 1 in equation (1).

$y=1(x+4)^2-1$

Therefore, the The correct option is second option, i.e., $y=(x+4)^2-1$.