a quadratic function has 3 characteristics x=-1 is the equation for the axis of symmetry... 3 is the x intercept ...y=32 is the maximum value
x=-1 is the equation for the axis of symmetry
y = a(x+1)^2 + k
y=32 is the maximum value
so a is negative
y = -a(x+1)^2 + 32
3 is the x intercept
This cannot be the only x-intercept, or the maximum value would be 0. So, assuming y(3) = 0,
-a(4^2)+32 = 0
x = 2
y = -2(x+1)^2 + 32
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To find the equation of a quadratic function with these characteristics, we can use the vertex form of a quadratic function. The vertex form of a quadratic function is given by:
y = a(x - h)^2 + k
where (h, k) is the vertex of the parabola.
Given that the axis of symmetry is x = -1, we know that the vertex of the parabola lies on this line. Therefore, the x-coordinate of the vertex, h, is -1.
Also, given that the x-intercept is 3, we know that when the parabola intersects the x-axis, the y-coordinate is 0. Substituting these values into the vertex form, we have:
0 = a(3 - (-1))^2 + k
0 = a(4)^2 + k
0 = 16a + k
We also know that y = 32 is the maximum value. Since the vertex form represents a parabola that opens upwards, the vertex height, k, represents the maximum value. Therefore, k = 32.
Substituting this information into the equation we obtained earlier, we have:
0 = 16a + 32
Simplifying the equation:
16a = -32
a = -2
Now we have the value of a, which allows us to determine the equation of the quadratic function. Substituting the values of h, k, and a into the vertex form equation, we get:
y = -2(x - (-1))^2 + 32
Simplifying:
y = -2(x + 1)^2 + 32
This is the equation for the quadratic function that satisfies the given characteristics.