Write a cubic function (in standard form) whose graph passes through given points: (-4), (0,10), (2,0) ,(5,0)
The two roots of 2 and 5 mean
y = a(x-2)(x-5)(x-h)
for some h.
Since y(0)=10,
a(-2)(-5)(-h) = 10
ah = -1
Now, fix the bogus (-4) point and see what a and h have to be.
To find a cubic function in standard form that passes through the given points, we'll use the general form of a cubic function: f(x) = ax^3 + bx^2 + cx + d.
Let's start by substituting the x and y values of the first point, (-4), into the function:
-4: a(-4)^3 + b(-4)^2 + c(-4) + d = -4a + 16b - 4c + d
Since the only information we have is the x-value, we can't determine the value of this term. Therefore, we'll assume the y-coordinate is 0 to satisfy this point.
Substituting the values for the second point, (0, 10), into the function:
0: a(0)^3 + b(0)^2 + c(0) + d = 0a + 0b + 0c + d = d
We know d is equal to 10 because the y-coordinate of the second point is 10. So, now we have:
f(x) = -4a + 16b - 4c + 10
Let's move on to the third point, (2, 0):
2: a(2)^3 + b(2)^2 + c(2) + d = 8a + 4b + 2c + d = 0
Similarly, using the fourth point, (5, 0):
5: a(5)^3 + b(5)^2 + c(5) + d = 125a + 25b + 5c + d = 0
Now, we have a system of linear equations with four variables (a, b, c, and d). To solve this system, we can use any method such as substitution, elimination, or matrix calculations.
Here's how you can solve this system of equations using the substitution method:
1. Solve the second equation for d: d = 0
2. Substitute d = 0 in the first equation: 8a + 4b + 2c + 0 = 0
3. Simplify: 8a + 4b + 2c = 0
4. Substitute d = 0 in the third equation: 125a + 25b + 5c + 0 = 0
5. Simplify: 125a + 25b + 5c = 0
Now, we have two equations with three variables (a, b, and c). We can solve this using the same process. I will continue with the substitution method:
6. Solve the first equation for a: a = -(4b + 2c)/8 = -(2b + c)/4
7. Substitute this value of a in the second equation: 125(-(2b + c)/4) + 25b + 5c = 0
8. Simplify: -50b - 12.5c + 25b + 5c = 0
9. Combine like terms: -25b - 7.5c = 0
Now, we have one equation with two variables (b and c). Let's solve for one variable:
10. Solve the equation for b: b = (-0.5c)/(-25) = 0.02c
11. Substitute this value of b in the equation: -25(0.02c) - 7.5c = 0
12. Simplify: -0.5c - 7.5c = 0
13. Combine like terms: -8c = 0
14. Solve for c: c = 0
Now we know that c = 0. Substituting this back into the equation b = 0.02c, we get b = 0. Multiplying the a term, -(2b + c)/4, by 4, we find a = 0.
Finally, substituting a = 0, b = 0, c = 0, and d = 10 into the cubic function f(x) = -4a + 16b - 4c + 10, we obtain:
f(x) = 10
Therefore, the cubic function in standard form that passes through the given points is f(x) = 10.