Find an equation of quadratic function the graph passing through the pointers A(2,3) B(-1,0) C(1,0) and draw the parabola
We can use the fact that a quadratic function can be written in the form:
y = ax^2 + bx + c
where x and y are the variables, and a, b, and c are constants.
To find these constants, we can use the given points.
Using point A(2,3), we get:
3 = a(2)^2 + b(2) + c
3 = 4a + 2b + c (equation 1)
Using point B(-1,0), we get:
0 = a(-1)^2 + b(-1) + c
0 = a - b + c (equation 2)
Using point C(1,0), we get:
0 = a(1)^2 + b(1) + c
0 = a + b + c (equation 3)
Now we have three equations with three unknowns (a, b, and c). We can solve this system of equations using substitution or elimination methods. Here we will use elimination.
Subtracting equation 2 from equation 1, we get:
3 = 4a + 2b + c
-0 = -a + b - c
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3 = 3a + 3b (equation 4)
Subtracting equation 3 from equation 2, we get:
0 = a - b + c
-0 = a + b + c
---------------
0 = 2a (equation 5)
Now we can solve for a:
equation 5 gives: 0 = 2a => a = 0
Substituting a = 0 into equation 4, we get:
3 = 3b => b = 1
Substituting a = 0 and b = 1 into equation 1, we get:
3 = 0(2)^2 + 1(2) + c => c = 1
Therefore, the equation of the quadratic function is:
y = x+1
To draw the parabola, we can use the vertex form of the equation:
y = a(x - h)^2 + k
where (h,k) is the vertex of the parabola, and a is the same constant as before (in this case, a = 0).
The vertex is at the point (-b/2a, c - b^2/4a), which we can find using the coefficients we just found:
vertex = (-1/(2*0), 1 - 1/(4*0)) = (undefined, 1)
This means that the parabola is a vertical line passing through the point (undefined, 1). We can also see this from the equation y = x + 1, which is a linear function with slope 1 and y-intercept 1.
Here is a graph of the three given points and the line y = x + 1:
To find the equation of a quadratic function that passes through the given points A(2,3), B(-1,0), and C(1,0), we can start by using the standard form of a quadratic function:
y = ax^2 + bx + c
We can substitute the x and y coordinates of the points A, B, and C into this equation to create a system of equations.
For point A(2,3):
3 = a(2)^2 + b(2) + c
For point B(-1,0):
0 = a(-1)^2 + b(-1) + c
For point C(1,0):
0 = a(1)^2 + b(1) + c
Now, we can solve this system of equations to find the values of a, b, and c.
From the equation for point B, we can write c in terms of a and b:
c = -a - b
Substituting this expression for c in the other two equations, we get:
3 = 4a + 2b - a - b
0 = a + b - a - b
Simplifying these equations further, we have:
3 = 3a + b
0 = 0
Since the last equation is always true, it does not provide any additional information. We can ignore it and focus on the first equation:
3 = 3a + b
Now, we can solve this equation for b in terms of a:
b = 3 - 3a
Substituting this expression for b in the equation for point A, we get:
3 = 4a + 2(3 - 3a) - a
3 = 4a + 6 - 6a - a
3 = -3a + 6
-3 = -3a
a = 1
Now that we have found the value of a, we can substitute it back into the expression we found for b:
b = 3 - 3(1)
b = 0
Finally, let's substitute the values of a and b back into the expression for c:
c = -a - b
c = -(1) - 0
c = -1
Therefore, the equation of the quadratic function that passes through the given points is:
y = x^2 + 0x - 1
Now, let's graph this equation to show the parabola:
Here is the graph of the parabola:
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Note: This is a rough sketch of the parabola, and it may not be perfectly accurate.