find an equation of quadratic function the graph passes through the points A(2,3)B(-1,0) and C (1,0)
To find the quadratic function that passes through these three points, we can use the general form of a quadratic function:
f(x) = ax^2 + bx + c
where a, b, and c are unknown constants that we need to find. To do this, we can plug in the coordinates of each point and get a system of three equations:
f(2) = 4a + 2b + c = 3
f(-1) = a - b + c = 0
f(1) = a + b + c = 0
We can solve this system of equations by eliminating one of the unknowns, say c. For example, we can add the second and third equations to get:
2a = 0
which implies that a = 0. Substituting this into the other two equations, we can solve for b and c:
b = 3/2
c = -3/4
Therefore, the quadratic function is:
f(x) = (3/2)x - 3/4
or
f(x) = 1.5x - 0.75 (in decimal form)
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 use the standard form of a quadratic function: y = ax^2 + bx + c.
Step 1: Plug in the coordinates of point A(2, 3) into the equation:
3 = a(2)^2 + b(2) + c
9 = 4a + 2b + c (Equation 1)
Step 2: Plug in the coordinates of point B(-1, 0) into the equation:
0 = a(-1)^2 + b(-1) + c
0 = a - b + c (Equation 2)
Step 3: Plug in the coordinates of point C(1, 0) into the equation:
0 = a(1)^2 + b(1) + c
0 = a + b + c (Equation 3)
Now we have a system of three equations with three unknowns (a, b, c). We can solve this system of equations to find the values of a, b, and c.
Subtract Equation 2 from Equation 3:
0 = (a + b + c) - (a - b + c)
0 = 2b
b = 0
Substitute b = 0 into Equation 2:
0 = a - 0 + c
0 = a + c (Equation 4)
Substitute b = 0 into Equation 1:
9 = 4a + 0 + c
9 = 4a + c (Equation 5)
Combine Equations 4 and 5:
0 + 9 = a + c + 4a
9 = 5a + c (Equation 6)
Since we have the values of a = 0 and b = 0, substitute these values into the standard form of quadratic function y = ax^2 + bx + c:
y = 0x^2 + 0x + c
y = c
Now we know that c = y, we substitute c = 9 into Equation 6 and solve for a:
9 = 5a + 9
5a = 0
a = 0
Substituting the values of a = 0 and b = 0 into the standard form of quadratic function y = ax^2 + bx +c:
y = 0x^2 + 0x + 9
y = 9
Therefore, the equation of the quadratic function that passes through the points A(2, 3), B(-1, 0), and C(1, 0) is y = 9.