find the equation of the cubic equation show graph passes through the points (3,0) and (1,4) and is tangent to the axis at the origin???
I honestly cannot figure this out and can you explain the graph to me too?
Since it is tangent at the origin, there has to be double root at (0,0). Also (3,0) is an x-intercept
so we can let the equation be
y = ax^2(x-3)
but (1,4) lies on it
4 = a(1)(-2)
a = -2
function is
f(x) = -2x^2(x-3)
expand if needed
Proof:
http://www.wolframalpha.com/input/?i=plot+-2x%5E2%28x-3%29+%2C+x+%3D+-2%2C+x+%3D+4
To find the equation of a cubic equation that passes through the points (3,0) and (1,4) and is tangent to the axis at the origin, we can follow these steps:
Step 1: Determine the equation of the cubic equation in the form of y = ax^3 + bx^2 + cx + d.
Step 2: Apply the given conditions to determine the values of a, b, c, and d.
Step 3: Substitute these values into the equation to find the final equation of the cubic function.
Now, let's go through the steps in detail.
Step 1: Determine the equation of the cubic equation in the form of y = ax^3 + bx^2 + cx + d.
The general form of a cubic equation is y = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
Step 2: Apply the given conditions to determine the values of a, b, c, and d.
Condition 1: The cubic equation passes through the point (3,0).
Substituting x = 3 and y = 0 in the equation, we get:
0 = 27a + 9b + 3c + d --(1)
Condition 2: The cubic equation passes through the point (1,4).
Substituting x = 1 and y = 4 in the equation, we get:
4 = a + b + c + d --(2)
Condition 3: The cubic equation is tangent to the x-axis at the origin.
When the cubic equation is tangent to the x-axis at the origin, it means that the origin (0,0) is a turning point. This implies that the slope of the equation at the origin is zero. Find dy/dx and substitute x = 0 to obtain the condition:
0 = 3a(0)^2 + 2b(0) + c --(3)
Step 3: Solve the system of equations formed by (1), (2), and (3) to find the values of a, b, c, and d.
Solving these equations will give us the values of a, b, c, and d. Once we have these values, we can substitute them back into the equation in Step 1 to obtain the final equation of the cubic function.
Now, as for explaining the graph of the cubic equation, it is a curve that can take various shapes depending on the values of a, b, c, and d. A cubic function typically has one or two local maximum or minimum points and can have a positive or negative leading coefficient (a).
In this case, since the cubic equation is tangent to the x-axis at the origin, it means the graph will touch the x-axis at the origin, and the origin will be a turning point.
I hope this explanation helps you understand the process of finding the equation of a cubic equation and the graph of it.