how do you determine the the quadratic function if the problem gives you a graph with the line has points of (-3,5) and (0,-4)?
Two points only is not enough to establish a quadratic, unless one of those points is the vertex.
If one of the points is a vertex, use the method I just showed you in your previous post.
Let me know what you think.
the (-3,5) is the vertex how do i use that i am not completely understand
where do i plug in thoes number
Look back at my explanation in the previous post
for vertex (h,k), we get
y = a(x-h)^2 + k
so for vertex (-3,5), we get
y = a(x-(-3) )^2 + 5
or
y = a(x + 3)^2 + 5
Now plug in the other point (0, -4)
-4 = a(0+3)^2 + 5
-4 = a(9) + 5
-9 = 9a
a = -1
y = -(x+3)^2 + 5
how did you get the nine after the A?
mmmhhh?
(0+3)^2
= 3^2
= 9
To determine the quadratic function from a graph, you need to identify the vertex of the parabola, which represents the minimum or maximum point, and the roots or x-intercepts of the quadratic equation.
In this case, the given graph has two points (-3,5) and (0,-4). The point (-3,5) lies on the parabola, so we can substitute these values into the general quadratic equation.
The general form of a quadratic function is:
y = ax^2 + bx + c
Using the point (-3,5), we substitute the x-coordinate (-3) and y-coordinate (5) into the equation:
5 = a(-3)^2 + b(-3) + c
which simplifies to:
5 = 9a - 3b + c
Similarly, using the point (0,-4), we substitute the x-coordinate (0) and y-coordinate (-4) into the equation:
-4 = a(0)^2 + b(0) + c
-4 = c
So, we have obtained the value of c as -4.
Substituting this c-value into our previous equation (5 = 9a - 3b + c), we get:
5 = 9a - 3b - 4
Now, we have two equations:
5 = 9a - 3b - 4
-4 = c
Simplifying the first equation:
9a - 3b = 5 + 4
9a - 3b = 9
Dividing throughout by 3:
3a - b = 3
Now, we have a system of equations:
3a - b = 3
-4 = c
To solve this system, we can choose any method of solving simultaneous equations, such as substitution or elimination. Let's use substitution:
Rearrange the first equation to solve for b:
b = 3a - 3
Substitute this value of b into the equation c = -4:
-4 = -4
Since -4 = -4 is a true statement, any value of a will give us the solution. Let's choose a = 1 for simplicity.
Substituting a = 1 into the equation b = 3a - 3, we get:
b = 3(1) - 3
b = 0
So, we have obtained the values of a = 1, b = 0, and c = -4.
Therefore, the quadratic function represented by the given graph is:
y = x^2 - 4