Determine the equation of a quadratic function that satisfies each set of conditions.
a) x-intercepts 1 and -1, y-intercept 3
b) x-intercept 3, and passing through the point (1, -2)
c) x-intercepts -1/2 and 2, y-intercept 4
Please explain your answers.
b) can be satisified by 2x^2-3x+3=y but it may be satisfied by other equations.
To determine the equation of a quadratic function given certain conditions, we can use the general form of a quadratic equation: y = ax^2 + bx + c, where a, b, and c are constants.
a) x-intercepts 1 and -1, y-intercept 3:
The x-intercepts of a quadratic function occur when y = 0. Therefore, we have the following two equations:
1) 0 = a(1)^2 + b(1) + c
2) 0 = a(-1)^2 + b(-1) + c
In addition, we are given the y-intercept, which is the point (0, c). So, we have the equation:
3) 3 = a(0)^2 + b(0) + c
Now, we can solve this system of equations to find the values of a, b, and c.
b) x-intercept 3, passing through the point (1, -2):
Again, the x-intercept occurs when y = 0. So, we have the equation:
1) 0 = a(3)^2 + b(3) + c
The equation for a point on the graph, (1, -2), is:
2) -2 = a(1)^2 + b(1) + c
We can solve this system of equations to find the values of a, b, and c.
c) x-intercepts -1/2 and 2, y-intercept 4:
The x-intercepts occur when y = 0, so we have the following two equations:
1) 0 = a(-1/2)^2 + b(-1/2) + c
2) 0 = a(2)^2 + b(2) + c
We also know the y-intercept, which is the point (0, c). So, the equation becomes:
3) 4 = a(0)^2 + b(0) + c
We can solve this system of equations to find the values of a, b, and c.
By solving these systems of equations, we can determine the values of a, b, and c, which will allow us to write the equations of the quadratic functions that satisfy the given conditions.
a) start with
y = a(x-1)(x+1), then put (0,3) into that to find a
c) do it the same as a)
b) not enough information unless your point (1,-2) is the vertex.