Determine the general equation for the family of quartic functions having zeros at x=-3/2, x=0, x=1/2 and x=2. Determine an equation for a family;y member whose graph passes through the post (-1,4.5)
y = a(2x+3)(x)(2x-1)(x-2)
now you can plug in the point to find a
To determine the general equation for the family of quartic functions with zeros at x = -3/2, x = 0, x = 1/2, and x = 2, we need to use the fact that a quartic equation can be expressed as a product of linear factors corresponding to its zeros.
The equation can be written as:
f(x) = a(x + 3/2)(x)(x - 1/2)(x - 2)
To find the specific equation for a member of this family that passes through the point (-1, 4.5), we will substitute the x and y values into the equation and solve for a.
Plugging in x = -1 and y = 4.5, we have:
4.5 = a(-1 + 3/2)(-1)(-1 - 1/2)(-1 - 2)
Simplifying further:
4.5 = a(1/2)(-1)(-3/2)(-3)
4.5 = a(1/2)(-1)(-9/2)(-3)
4.5 = a(27/2)
To solve for a, we can divide both sides by 27/2:
4.5 / (27/2) = a
Multiplying by the reciprocal:
4.5 * (2/27) = a
This simplifies to:
a = 0.1667 (rounded to four decimal places)
Therefore, the equation for a family member that satisfies the given conditions is:
f(x) = 0.1667(x + 3/2)(x)(x - 1/2)(x - 2)
To determine the general equation for the family of quartic functions, we need to consider the zeros provided. A quartic function has the general form:
f(x) = a(x - r)(x - s)(x - t)(x - u)
where a is a constant, and r, s, t, and u are the zeros.
In this case, the zeros are x = -3/2, x = 0, x = 1/2, and x = 2. So the equation becomes:
f(x) = a(x + 3/2)(x)(x - 1/2)(x - 2)
Next, we need to determine the value of 'a'. To find this, we can use the given point (-1, 4.5) that the graph passes through.
Substitute x = -1 and f(x) = 4.5 into the equation:
4.5 = a(-1 + 3/2)(-1)(-1 - 1/2)(-1 - 2)
Simplifying,
4.5 = a(-1/2)(-1)(-3/2)(-3)
Now, simplify the expression inside the brackets:
4.5 = a(1/2)(-1)(3/2)(-3)
Multiply all the numbers together:
4.5 = a(9/4)
Now, solve for 'a' by dividing both sides by (9/4):
a = (4/9) * 4.5
a = 2
Therefore, the general equation for the family of quartic functions with zeros at x = -3/2, x = 0, x = 1/2, and x = 2 is:
f(x) = 2(x + 3/2)(x)(x - 1/2)(x - 2)
And, the equation for a specific member of this family, whose graph passes through the point (-1, 4.5), is:
f(x) = 2(x + 3/2)(x)(x - 1/2)(x - 2)